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■<JI/J  I  I  V.(    J 


THE    ADJUSTMENT 


OF 


OBSERVATIONS 


BV  THE  METHOD    OF  LEAST  SQUARES    WITH 
APPLICATIONS   TO    GEODETIC  WORK 


BY 

THOMAS  WALLACE  WRIGHT,  M.A.,C.E. 

Professor   Emeritus,  Union   College  ;   Formerly  Assistant    Engineer   Survey   of  the 

NORTJIERN    AND    NORTHWESTERN    LaKES 


WITH    THE    COOPERATION    OF 


JOHN  FILLMORE   HAYFORD,  C.E. 

Chief  op  the  Computing    Division  and  Inspector  of    Geodetic  Work,  U.  S. 
Coast  and  Geodftic  Survey 


SECOND    EDITION 


NEW   YORK 
D.  VAN    NOSTRAND    COMPANY 

1 906  ^^ 


Copyright,  1906 
T.   W.   WRIGHT 


Stanbopc  press 

r.     H.     GILSON      COMPANV 
BOSTON,      U.S.A. 


PREFACE 


This  book  originated  in  this  way.  While  employed  as 
Assistant  Engineer  on  the  Survey  of  the  Northern  and  North- 
western Lakes,  many  questions  came  up  in  the  course  of  the 
work  for  which  no  help  could  be  found  in  any  publication  in 
the  library  of  the  Survey.  Conclusions  were,  in  general,  reached 
often  after  long  continued  discussions.  I  at  the  time  made 
notes  of  the  questions  and  of  the  solutions  obtained,  in  order 
that  if  similiar  questions  should  again  come  up  they  might  more 
readily  be  dealt  with. 

At  the  close  of  the  Survey,  I  had  a  large  collection  of  notes 
of  this  kind.  Shortly  afterwards,  on  entering  college  work, 
these  notes  were  arranged  in  systematic  order.  Also  at  the 
same  time  an  account  of  the  Coast  and  Geodetic  Survey  methods 
of  work  was  added. 

As  only  a  comparatively  small  edition  was  printed  from  type 
in  the  first  place,  the  book  has  been  out  of  print  for  a  number 
of  years,  though  repeated  requests  have  been  made  for  copies. 
In  the  spring  of  1903,  Superintendent  Tittmann,  U.  S.  Coast 
and  Geodetic  Survey,  wrote  to  the  publishers  as  follows  :  "As 
this  book  is  one  of  exceeding  importance  to  the  Survey,  and 
will  grow  even  more  needful  in  this  work,  and,  as  I  take  it,  in 
many  fields  of  scientific  engineering  operations,  I  beg  to  inquire 
whether  you  anticipate  issuing  a  new  edition  of  this  useful 
book.?" 

This  led  to  some  correspondence,  and  it  was  finally  arranged 
that  Mr.  John  F.  Hayford,  Chief  of  the  Computing  Division 
and  In.spector  of  Geodetic  Work,  should  assist  in  revising  the 
book.      Most  of  the  new  matter  has  been   contributed  by  Mr. 


4 164 34 


iv  PREFACE 

Hayford.  Indeed,  so  important  have  been  his  contributions 
that  I  have  insisted  that  his  name  appear  on  the  title  page. 

We  have  been  assisted  by  the  leading  members  of  the  Com- 
puting Division,  notably  by  Mr.  M.  H.  Doolittle.  For  Chapters 
VII  and  IX,  Mr.  Hayford  is  alone  responsible.  Chapter  VII 
contains  an  authoritative  account  of  the  latest  methods  in  use 
in  the  Coast  Survey  for  the  adjustment  of  a  triangulation.  The 
account  given  in  the  first  edition  was,  at  the  time  of  its  publi- 
cation, nearly  correct.  But  during  the  last  twenty  years  rapid 
progress  has  been  made  in  the  Computing  Division  of  the 
C.  &  G.  S.  in  improving  methods  of  computation,  under  the 
direction  of  Mr.  C.  A.  Schott  and  his  successor  Mr.  Hayford, 
with  the  able  assistance  of  Mr.  Doolittle,  Mr.  E.  H.  Courtenay, 
Mr.  C.  H.  Kummel,  Mr.  A.  L.  Baldwin,  and  others,  who  have 
done  original  thinking.  The  changes  are  the  result  of  seven 
decades  of  experience  in  the  Survey. 

Chapter  IX  contains  an  exceedingly  important  application 
of  the  method  of  least-squares  to  the  selection  of  methods  of 
observation. 

The  leading  principles  that  have  been  followed  in  making 
the  revision  are  these  : 

1.  Matter  that  was  curious  only  and  without  application  has 
been  omitted. 

2.  Matter  relating  to  description  of  instruments  and  methods 
of  observation  is  in  general  eliminated. 

3.  Statements  of  formulas  not  pertaining  to  least-squares  are 
omitted. 

4.  Following  American  custom,  the  term  "probable  error" 
is  used  instead  of  "mean-square  error." 

5.  In  order  not  to  increase  the  size  of  the  book  all  applica- 
tions to  Physics,  etc.,  have  been  omitted. 

The  number  of  pages  has  been  cut  down  from  four  hundred 
and  thirty-seven  to  303. 

T.  W.  \V. 
Schenectady,  Nov.,  1905. 


TABLE    OF    CONTENTS 


CHAPTER   I 
Introduction 

The  instrument i 

Extei  nal  conditions   4 

The  observer 5 

CHAPTER    II 

The  Law  of  Error 

The  arithmetic  mean 9 

The  arithmetic  mean  the  most  plausible  value 11 

When  the  arithmetic  mean  gives  the  true  value 12 

Sum  of  residuals  is  zero  12 

Sum  of  squares  of  residuals  a  minimum 13 

Law  of  error  of  a  single  observed  quantity 13 

Derivation  of  law  of  distribution  of  errors 13 

Second  derivation,  on  Hagen's  hypothesis 16 

The  principle  of  least  squares 19 

Law    of    error    of    a    linear     function    of    independently    observed 

quantities 20 

Comparison  of  the  accuracy  of  different  series  of  observations 22 

The  mean-square  error 22 

The  probable  error 22 

The  average  error 24 

The  probability  curve 27 

The  law  of  error  applied  to  an  actual  series  of  observations 30 

Effect  of  extending  the  limits  of  error  to  J:;  co 30 

General  conclusions 33 

Classification  of  observations 33 

CHAPTER    III 

Adjustment  of  Direct  Observations  of  One  Unknown 

Observed  values  of  equal  quality 35 

The  most  probable  value  —  The  arithmetic  mean 35 

Control  of  the  arithmetic  mean 3f> 

Precision  of  the  arithmetic  mean , 38 

V 


vi  TABLE   OF   CONTENTS 

PAGE 

Bessel's  formula 3^ 

Distinction  between  residuals  and  errors 40 

Peters'  formula 4° 

Control  of  [7/=] 42 

Approximate  method  of  finding  precision 42 

The  law  of  error  tested  by  experience 44 

Cautions  as  to  tests  of  precision 45 

Systematic  errors    5 1 

Observed  values  of  different  quality  52 

The  most  probable  value  — The  weighted  mean 54 

Combining  weights   55 

Reduction  of  observations  to  a  common  standard 56 

Control  of  weighted  mean  57 

Precision  of  weighted  mean 58 

Observed  values  multiples  of  the  unknown 60 

Precision  of  a  linear  function  of  independently  observed  values 62 

Miscellaneous  examples 67 

Weighting  of  observations 74 

An  approximate  method 76 

Weighting  when  constant  error  is  present 77 

Assignment  of  weight  arbitrarily 82 

Combination  of  good  and  inferior  work 83 

The  weight  a  function  of  our  knowledge 84 

General  remarks 87 

Rejection  of  observations 87 

CHAPTER   IV 

Adjtistment  of  Indirect  Observations 

Determination  of  the  most  probable  values 93 

Formation  of  the  normal  equations 96 

Control  of  the  formation 100 

Forms  of  computing  the  normal  equations loi 

With  multiplication  tables  or  a  machine loi 

With  a  table  of  logarithms 102 

With  a  table  of  squares 103 

Solution  of  the  normal  equations 105 

The  method  of  substitution 106 

Controls  of  the  solution 107 

Forms  of  solution 109 

Solution  without  logarithms 109 

Solution  with  logarithms 112 

The  method  of  indirect  elimination . 114 

Doolittle  method  of  solution 114 

Precision  of  the  most  probable  (adjusted)  values 121 


TABLE  OF  CONTENTS  vii 

PAGE 

First  method  of  finding  the  weights 122 

Special  case  of  two  and  three  unknowns 125 

Modifications  of  the  general  method 127 

Second  method  of  finding  the  weights 129 

The  probable  error  of  a  single  observation 132 

Methods  of  computing  [v-] I33 

Precision  of  any  function  of  the  adjusted  values  (three  methods) 137 

Average  value  of  the  ratio  of  the  weight  of  an  observed  value  to  its  ad- 
justed value '-^3 

Two  special  artifices M4 

CHAPTER  V 

Adjusttnent  of  Condition  Observations 

General  statement ^49 

Direct  solution—  Method  of  independent  unknowns 150 

Indirect  solution  —  Method  of  correlates 152 

Precision  of  the  adjusted  values  or  of  any  function  of  them 158 

The  probable  error  of  an  observation  of  weight  unity 158 

Weight  of  the  function '^^ 

Solution  in  two  groups ^"3 

Program  of  solution ^7° 

Precision  of  the  adjusted  values  or  of  any  function  of  them 173 

Solution  by  successive  approximation ^77 

CHAPTER   VI 

Application  to   the  Adjustment  of  a   Triangulation  — Method  of  Angles 

General  statement '^° 

The  method  of  independent  angles '82 

The  local  adjustment '^5 

Number  of  local  equations ^^^ 

The  general  adjustment ^^° 

The  angle  equations '°9 

Number  of  angle  equations '9^ 

The  side  equations ^93 

Position  of  pole '95 

Reduction  to  the  linear  form '97 

Check  computation '9° 

Position  of  pole ^°'' 

Number  of  side  equations 2°- 

Check  of  the  total  number  of  conditions 202 

Manner  of  selecting  the  angle  and  side  equations 203 

Adjustment  of  a  quadrilateral 206 

Solution  by  independent  unknowns 207 

Precision  of  the  adjusted  values 208 


viii  TABLE  OF  CONTENTS 

PAGE 

Solution  by  correlates 210 

Precision  of  the  adjusted  values 211 

Solution  in  two  groups 213 

Precision  of  the  adjusted  values 218 

Solution  by  groups 223 

The  local  adjustment 227 

The  general  adjustment 227 

Adjustment  of  a  quadrilateral  —  Approximate  method 228 

Adjustment  of  a  quadrilateral  —  Rigorous  method 231 

Adjustment  of  a  single  triangle 233 

Adjustment  of  a  central  polygon 234 

Approximate  method  of  tinding  the  precision 237 

CHAPTER  VII 

Application  to  the  Adjustvie/it  of  a  Triangiilation  — Method  of 
Directions 

General  statement 239 

Local  adjustment 243 

Figure  adjustment 243 

Condition  equations 244 

Correlate  equations 246 

Normal  equations 246 

The  best  side  equations 247 

Length,  azimuth,  latitude  and  longitude  condition  equations 250 

Length  condition  equations 252 

Azimuth  condition  equations 253 

Latitude  and  longitude  condition  equations 255 

Breaking  a  net  into  sections 259 

CHAPTER  Vni 

Application  to  Base-Line  Measicronent  and  to  Leveling 

Precision  of  base-line  measurement 260 

Precision  of  each  bar  length  the  same 263 

Precision  of  each  bar  length  not  necessarily  the  same 265 

Application  to  leveling 267 

Method  of  indirect  observations 268 

Method  of  conditioned  observations  268 

Assignment  of  weights 270 

CHAPTER    IX 

Application  to  Selection  of  Methods  of  Observation 

General  statement 272 


TABLE  OF  CONTENTS  ix 


PAGE 


Distinction  between  accidental,  systematic,  and  constant  errors 273 

More  accurate  definition  of  probable  error 274 

, Detection  of  systematic  and  constant  errors 276 

Zenith  telescope  latitude  observations 278 

Telegraphic  longitude  observations 283 

Other  illustrations 286 

APPENDIX 

Table  I,  Values  of  61  (/) 291 

Table  1 1,  Factors  for  Bessel's  probable  error  formula 292 

Table  III.  Factors  for  Peters'  probable  error  formula 293 


THE  ADJUSTMENT  OF  OBSERVATIONS 

CHAPTER    I 

INTRODUCTION 

The  factors  that  enter  into  the  measurement  of  a  quantity 
are,  the  observer,  the  instrument  employed,  and  the  conditions 
under  which  the  measurement  is  made. 

I.  The  Instrument. —  If  the  measure  of  a  quantity  is  deter- 
mined by  untrained  estimation  only,  the  result  is  of  little  value. 
The  many  external  influences  at  work  hinder  the  judgment  from 
deciding  correctly.  For  example,  if  we  compare  the  descrip- 
tions of  the  path  of  a  meteor  as  given  by  a  number  of  people 
who  saw  the  meteor  and  who  try  to  tell  what  they  saw,  it  would 
be  found  impossible  to  locate  the  path  satisfactorily.  The  work 
of  the  earlier  astronomers  was  of  this  vague  kind.  There  was 
no  way  of  testing  assertions,  and  theories  were  consequently 
plentiful. 

The  first  great  advance  in  the  science  of  observation  was  in 
the  introduction  of  instruments  to  aid  the  senses.  The  instru- 
ment confined  the  attention  of  the  observer  to  the  point  at  issue, 
and  helped  the  judgment  in  arriving  at  conclusions.  As  with  a 
rude  instrument  different  observers  would  get  the  same  result, 
it  is  not  to  be  wondered  at  that  for  a  long  time  a  single  instru- 
mental determination  was  considered  sufficient  to  give  the  value 
of  the  quantity  measured. 

The  next  advance  was  in  the  questioning  of  the  instrument 
and  in  showing  that  a  result  better  on  the  whole  than  a  single 
direct  measurement  could  be  found.  This  opened  the  way  for 
better  instruments  and  better  methods  of  observation.  For 
example,  Gascoigne's  introduction  of  cross-hairs  into  the  focus  of 

I 


2         THE  ADJUSTMENT  OF  OBSERVATIONS 

the  telescope  led  to  better  graduated  circles  and  to  better 
methods  of  reading  them,  resulting  finally  in  the  reading  micro- 
scopes now  almost  universally  used.  The  culminating  point  was 
reached  by  Bessel,  who,  by  his  systematic  and  thorough  investi- 
gation of  instrumental  corrections  and  methods  of  observation, 
may  be  said  to  have  almost  exhausted  the  subject.  He  confined 
himself,  it  is  true,  to  astronomical  and  geodetic  instruments,  but 
his  methods  are  of  universal  application. 

The  questioning  of  an  instrument  naturally  arises  from  noti- 
cing that  there  are  discrepancies  in  repeated  measurements  of  a 
magnitude  with  the  same  instrument,  or  in  measures  made  with 
different  instruments.  Thus,  if  a  distance  was  measured  with 
an  ordinary  chain,  and  then  measured  with  a  standard  whose 
length  had  been  very  carefully  determined,  and  the  two  measure- 
ments differed  widely,  we  should  suspect  the  chain  to  be  in  error, 
and  proceed  to  examine  it  before  further  measuring.  So,  dis- 
crepancies found  in  measurements  made  with  the  same  measure 
at  different  temperatures  have  shown  the  necessity  of  finding 
the  length  of  the  measure  at  some  fixed  temperature,  and  then 
applying  a  correction  for  the  length  at  the  temperature  at  which 
the  measurement  is  made. 

Corrections  to  directly  measured  values  are  thus  seen  to  be 
necessary,  and  to  be  due  to  both  internal  and  external  causes. 
The  internal  causes  arising  from  the  construction  of  the  instru- 
ment are  seen  to  be  in  great  measure  capable  of  elimination. 
From  geometrical  considerations  the  observer  can  tell  the 
arrangement  of  parts  demanded  by  a  perfect  instrument.  He 
can  compute  the  errors  that  would  be  introduced  by  certain  sup- 
posed irregularities  in  form  and  changes  of  condition.  The 
instrument-maker  cannot,  it  is  true,  fulfill  the  conditions  neces- 
sary for  a  perfect  instrument,  but  he  has  been  gradually 
approaching  them  more  and  more  closely.  It  is  to  be  remem- 
bered that,  even  if  an  instrument  could  be  made  perfect  at  any 
instant,  it  would  not  remain  so  for  any  great  length  of  time. 
It  hence  followed  as  the  next  great  advance  that  the  instru- 


INTRODUCTION  3 

ment  was  made  adjustable  in  most  of  its  parts,  so  that  the 
relative  positions  of  the  parts  are  under  the  control  of  the 
observer.  Tliis  is  getting  to  be  more  and  more  the  case  with 
the  better  class  of  instruments. 

2.  Not  only  is  error  diminished  by  the  improved  construction 
of  the  instrument,  but  also  by  more  refined  methods  of  handling 
it.  It  may  be,  indeed,  that  some  contrivances  beyond  those 
required  to  make  necessary  readings  for  the  measure  of  the 
quantity  in  question  may  be  needed.  Thus,  with  a  graduated 
circle,  regular  or  periodic  errors  of  graduation  may  be  expected. 
If  the  angle  between  two  signals  were  read  with  a  theodolite, 
the  reading  on  each  signal,  and  consequent  value  of  the  angle, 
would  be  influenced  by  the  periodic  errors  of  the  circle  of  the 
instrument.  Though  a  single  vernier  or  microscope  would  suffice 
to  read  the  circle  when  the  telescope  is  directed  to  the  signals, 
yet,  as  the  circle  is  incapable  of  adjustment,  we  can  only  get  rid 
of  the  influence  of  the  periodicity  by  employing  a  number  of  ver- 
niers or  microscopes  placed  at  equal  intervals  around  the  circle. 
It  happens  that  this  same  addition  of  microscopes  eliminates 
eccentricity  of  the  graduated  circle  as  well. 

This  same  principle  of  making  the  method  of  observation 
eliminate  the  instrumental  errors  is  carried  through  even  after 
the  nicest  adjustments  have  been  made.  Thus,  in  ordinary 
leveling,  if  the  backsights  and  foresights  are  taken  exactly  equal 
the  instrumental  adjustment  may  be  poor  and  still  good  work 
may  be  done.  But  good  work  is  more  likely  if  the  adjustments 
have  been  carefully  made,  as  if  for  unequal  sights,  and  still  the 
sights  are  taken  equal. 

Simplicity  of  construction  in  an  instrument  is  also  to  be  aimed 
at.  An  instrument  that  theoretically  ought  to  work  perfectly  is 
often  a  great  disappointment  in  practice.  A  striking  example  is 
the  compensating  base  apparatus  which  has  been  abandoned  on 
all  the  leading  surveys.  The  mechanical  and  thermal  difficulties 
have  proved  to  be  insurmountable,  and  the  compensating  bars 
iiave  been  replaced  by  others  of    much  sim])ler    construction. 


4  THE    ADJUSTMENT    OF    OBSERVATIONS 

Similarly,  the  repeating  theodolite  has  fallen  far  short  of  the 
expectations  of  its  first  advocates,  who  hoped  that  with  it  the 
errors  of  measurement  of  an  angle  could  be  reduced  almost 
indefinitely.  The  mechanical  difficulties  have  proved  insur- 
mountable, and  the  repeating  theodolite  is  now  known  to  be 
capable  of  no  greater  accuracy  than  the  direction  instrument. 

Such  is  the  perfection  at  present  attained  in  the  construction 
of  mathematical  instruments,  and  the  skill  with  which  they  can 
be  manipulated,  that  comparatively  little  trouble  in  making  ob- 
servations arises  from  the  instrument  itself. 

3.  External  Conditions.  —  The  great  obstacles  to  accurate 
work  arise  from  the  influence  of  external  conditions  —  condi- 
tions wholly  beyond  the  observer's  or  instrument-maker's  con- 
trol, and  whose  effect  can,  in  general,  neither  be  satisfactorily 
computed  nor  certainly  eliminated  by  the  method  of  observa- 
tion. We  have  no  means  of  finding  the  complex  laws  of  their 
action.  Many  of  them  can  be  avoided  by  not  observing  while 
they  operate  in  any  marked  degree.  Thus,  if  while  an  observer 
is  reading  horizontal  angles  on  a  high  tower  a  strong  wind 
arises,  it  may  be  necessary  for  him  to  stop  work.  If  the  air 
commences  to  "boil  "  with  extreme  violence,  it  may  also  be  best 
to  stop  work.  If  the  sun  shines  on  one  side  of  his  instru- 
ment, its  adjustments  would  be  so  disturbed  that  good  work 
could  not  be  expected.  So  in  comparisons  of  standards. 
Comparisons  made  in  a  room  subject  to  the  temperature  vari- 
ations of  the  outside  air  would  be  of  little  value.  The  standards 
should  not  only  not  be  exposed  to  sudden  temperature  changes 
during  comparisons,  but  at  no  other  time ;  for  it  has  been 
shown  that  the  same  standard  may  have  different  lengths  at 
the  same  temperature  after  exposure  to  wide  ranges  of  tem- 
perature. 

The  effects  of  external  disturbances  may  sometimes  be  elim- 
inated, in  part  at  least,  by  the  method  of  observation.  In  the 
measurement  of  horizontal  angles  where  the  instrument  is 
placed   on  a   high  tower,  the   influence  of  the  sun  causes  the 


INTRODUCTION  5 

center  post  or  tripod  of  the  station  to  twist  in  one  direction 
during  the  day.  When  this  influence  is  removed  at  night,  the 
twist  is  in  tlie  opposite  direction.  Assuming  the  twist  to  act 
uniformly,  its  effect  on  the  resuhs  is  eliminated  by  taking  the 
mean  of  the  readings  on  the  signals  observed  in  order  of 
azimuth,  and  then  immediately  in  the  rev^erse  order. 

Atmospheric  refraction  is  another  case  in  point.  In  observ- 
ing for  time  with  a  sextant,  the  effect  of  refraction  is  often 
eliminated  when  the  highest  degree  of  accuracy  is  required,  by 
taking  two  sets  of  observations  of  the  sun  at  about  the  same 
altitude,  one  before  and  the  other  after  noon.  On  the  other 
hand,  in  the  measurement  of  horizontal  angles,  if  long  lines  are 
sighted  over,  or  lines  passing  from  land  over  large  bodies  of 
water,  or  over  a  country  much  broken,  the  effects  of  re- 
fraction are  apt  to  be  very  marked.  As  we  have  no  means  of 
eliminating  the  discordances  arising  in  this  way  by  the  method 
of  observation,  all  we  can  do  is,  while  planning  a  triangulation, 
to  avoid  as  far  as  possible  the  introduction  of  such  lines. 

It  may  happen  that  the  effect  of  the  external  disturbances  on 
the  observations  can  be  computed  approximately  from  theoreti- 
cal considerations  assuming  a  certain  law  of  operation.  If  the 
correction  itself  is  small,  this  is  allowable.  As  an  example,  take 
the  zenith  telescope,  with  which  the  method  of  observing  for 
latitude  is  such  that  the  correction  for  refraction  is  so  small 
that  the  error  of  the  computed  value  is  not  likely  to  exceed 
other  errors  which  enter  into  the  work. 

4.  The  Observer.  —  Lastly,  we  come  to  the  observer  himself 
as  the  third  element  in  making  an  observation.  Like  the  ex- 
ternal conditions,  he  is  a  variable  factor  ;  all  new  observers  cer- 
tainly are. 

The  observer,  having  put  his  instrument  in  adjustment  and 
satisfied  himself  that  the  external  conditions  are  faxorable, 
should  not  begin  work  if  he  is  seeking  the  highest  degree  of 
accuracy  unless  he  considers  that  he  himself  is  in  liis  normal 
condition.     If    he    is   not   in   this   condition,   he  introduces  an 


6  THE    ADJUSTMENT    OF    OBSERVATIONS 

unknown  disturbing  element  unnecessarily.  He  is  also  more 
liable  to  make  mistakes  in  his  readings  and  in  his  record.  For 
the  same  reason  he  should  not  continue  a  series  of  observations 
too  long  at  one  time,  as  from  fatigue  the  latter  part  of  his  work 
will  not  compare  favorably  with  the  first.  In  time-determina- 
tions, for  instance,  nothing  is  gained  by  observing  from  dark 
until  daylight. 

The  observer  is  supposed  to  have  no  bias.  A  good  observer, 
having  taken  all  possible  precautions  with  the  adjustments  of 
his  instrument,  and  knowing  no  reason  for  not  doing  good  work, 
will  feel  a  certain  amount  of  indifference  towards  the  results 
obtained.  The  man  with  a  theory  to  substantiate  is  rarely  a 
good  observer,  unless,  indeed,  he  regards  his  theory  as  an 
enemy,  and  not  as  a  thing  to  be  fondled  and  petted. 

The  greater  an  observer's  experience,  the  more  do  his  habits 
of  observation  become  fixed,  and  the  more  mechanical  does  he 
become  in  certain  parts  of  his  work.  But  his  judgment  may  be 
constantly  at  fault.  Thus,  with  the  astronomical  transit  he  may 
estimate  the  time  of  a  star  crossing  a  wire  in  the  focus  of  the 
telescope  invariably  too  soon  or  invariably  too  late,  according  to 
the  nature  of  his  temperament.  If  he  is  doing  comparison 
work  involving  micrometer  bisections,  he  may  consider  the 
graduation  mark  sighted  at  to  be  exactly  between  the  center 
wires  of  the  microscope  when  it  is  constantly  on  the  same  side 
of  the  center.  This  fixed  peculiarity,  which  none  but  experi- 
enced observers  have,  is  known  as  their  personal  error,  ox  per- 
sonal equation. 

In  combining  one  observer's  results  with  those  of  another 
observer,  we  must  either  find  by  special  experiment  the  differ- 
ence of  their  personal  errors  and  apply  it  as  a  correction  to  the 
final  result,  or  else  eliminate  it  by  the  method  of  observation. 
Thus,  in  longitude  work  the  present  practice  is  to  eliminate  the 
effect  of  personal  error  from  the  final  result  by  having  the 
observers  change  places  at  the  middle  of  the  work. 

It  is  always  safer  to  eliminate  the  correction  by  the  method 


INTRODUCTION  7 

of  observing  rather  than  by  computing  for  it.  For  though  it 
may  happen  that  so  long  as  instruments  and  conditions  are  the 
same,  the  relative  personal  error  of  two  observers  may  be  con- 
stant, yet  some  apparently  trifling  change  of  conditions,  such, 
for  example,  as  illuminating  the  wires  of  the  instrument  differ- 
ently, may  cause  it  to  be  altogether  changed  in  character. 

On  account  of  personal  error,  if  for  no  other  reason,  it  is 
evident  that  no  number  of  sets  of  measures  obtained  in  the 
same  way  by  a  single  observer  ought  to  be  expected  to  furnish 
as  good  a  determination  of  the  value  of  a  quantity  as  might  be 
obtained  by  varying  the  form  of  making  the  observations  or 
increasing  the  number  of  observers. 

5.  When  all  known  corrections  for  instrument,  for  external 
conditions,  and  for  peculiarities  of  the  observer,  have  been 
applied  to  a  direct  measure,  have  we  obtained  a  correct  value  of 
the  quantity  measured  ?  That  we  cannot  say.  If  the  observa- 
tion is  repeated  a  number  of  times  with  equal  care,  different 
results  will  in  general  be  obtained. 

The  reason  why  the  different  measures  may  be  expected  to 
disagree  with  one  another  has  been  indicated  in  the  preceding 
pages.  There  may  have  been  no  change  in  the  conditions  of 
sufficient  importance  to  attract  the  observer's  attention  when 
making  the  observations,  but  he  may  have  handled  his  in- 
strument differently,  turned  certain  screws  with  a  more  or 
less  delicate  touch,  and  the  external  conditions  may  have  been 
different.  What  the  real  disturbing  causes  were,  he  has  no 
means  of  knowing  fully.  If  he  had,  he  could  correct  for  them, 
and  so  bring  the  measures  into  accordance.  Infinite  knowledge 
alone  could  do  this.  With  our  limited  powers,  we  must  e.xpect 
a  residuum  of  error  in  our  best  executed  measures,  and,  instead 
of  certainty  in  our  results,  look  only  for  probability. 

The  discrepancies  from  the  true  value  due  to  these  un- 
explained disturbing  causes  we  call  errors.  These  errors  are 
accidental,  being  wholly  beyond  all  our  efforts  to  control.  They 
are  as  likely  to  be  in  excess  as  in  defect.     If  we  can  ferret  out 


8         THE  ADJUSTMENT  OP  OBSERVATIONS 

the  law  of  their  operation,  they  cease  to  be  classed  as  errors,  and 
become  corrections. 

A  very  troublesome  source  of  discrepancies  in  measured 
values  arises  from  blunders  made  by  the  observer  in  reading  his 
instrument  or  in  recordmg  his  readings.  Blunders  from  imper- 
fect hearing,  from  transposition  of  figures  and  from  writing  one 
figure  when  another  is  intended,  from  mistaking  one  figure  on  a 
graduated  scale  for  another,  as  7  for  9,  3  for  8,  etc.,  are  not  un- 
common,—  nor  are  mistakes  of  level  reading  of  5  divisions,  of 
estimation  of  time  of  /*,  and  the  like. 

Carelessness  may  produce  the  same  effect  as  a  blunder  in 
reading  or  in  recording.  Thus,  handling  a  striding  level  roughly, 
or  bringing  a  heated  lamp  too  near  it,  may  affect  a  result  very 
seriously. 

Having,  then,  taken  all  possible  precautions  in  making  the 
observations,  and  applied  all  possible  corrections  to  the  observed 
values,  the  resulting  values,  which  we  shall  in  future  refer  to  as 
the  observed  values,  may  be  assumed  to  contain  only  accidental 
errors.  We  are,  then,  brought  face  to  face  with  the  question, 
How  shall  the  value  of  the  quantity  sought  be  found  from  these 
different  observed  values  ? 


CHAPTER    II 

THE    LAW    OF    ERROR 

TJie  Arit  June  tic  Mean 

6.  If  a  quantity  x  is  to  be  determined  by  measurement,  and 
M  is  a  measured  value  of  x,  then,  if  the  observation  were  per- 
fect, we  should  have 

X  —  M  =  o. 

But  since,  if  we  make  a  second  and  a  third  observation,  we  may 
not  find  the  same  value  as  we  did  at  first,  and  as  we  can  only 
account  for  the  difference  on  the  supposition  that  the  observa- 
tions are  not  perfect — that  is,  that  they  are  affected  with  cer- 
tain errors  —  we  should  rather  write 


X  —  3/2  =  ^2 

X-  M„  =  Ar. 


(0 


where  M^,  M.„  .  .  .  M„  are  the  observed  values,  and  Aj,  A^, 
.  .  .  A„  are  the  errors  of  the  observations. 

We  have  here  ;/  equations  and  ;/  -f-  i  unknowns.  W'liat 
principle  shall  we  call  to  our  aid  to  solve  these  equations  and 
so  find  X,  Aj,  A,„  .  .  .  A„.^  In  answering  this  question,  we 
shall  follow  the  order  of  natural  development  of  the  subject, 
which,  in  the  main,  is  also  the  order  of  its  historical  develop- 
ment. 

The  value  sought  must  be  some  function  of  the  observed 
values,  and  fall  between  the  largest  and  smallest  of  them. 
If  the  observed  values  are  arranged  according  to  their  magni- 
tudes, they  will  be  found  to  cluster   around  a  central  value. 

9 


lO  THE    ADJUSTMENT    OP    OBSERVATIONS 

On  first  thought,  the  value  that  would  be  chosen  as  the  value 
of  X  would  be  the  central  value  in  this  arrangement  if  the 
number  of  observations  were  odd,  and  either  of  the  two  central 
values  if  the  number  were  even.  In  other  words,  a  plausible 
value  of  the  unknown  would  be  that  observed  value  which  had 
as  many  observed  values  greater  than  it  as  it  had  less  than  it. 
Now,  since  a  small  change  in  any  of  the  observed  values,  other 
than  the  central  value,  would  in  general  produce  no  change  in 
the  result,  the  number  of  observations  remaining  the  same,  this 
method  of  proceeding  might  be  regarded  as  giving  a  plausible  re- 
sult, more  especially  if  the  observed  values  were  widely  discrepant. 

On  the  other  hand,  the  taking  of  the  central  value  is  objec- 
tionable, because  it  gives  the  preference  to  a  single  one  of  the 
observed  values  ;  while  if  these  values  are  supposed  to  be  equally 
worthy  of  confidence,  as  it  is  reasonable  to  take  them  in  the 
absence  of  all  knowledge  to  the  contrary,  each  ought  to  exert 
an  equal  influence  on  the  result.  We  may,  therefore,  with 
more  reason,  assume  the  value  of  x  to  be  a  symmetrical  function 
Xq  of  the  observed  values. 

The  simplest  symmetrical  function  of  the  observed  values 
that  can  be  chosen  as  the  form  for  x^^  is  their  arithmetic  mean 
—  that  is, 

n 

where  2  is  the  ordinary  algebraic  symbol  of  summation, 

[Af\ 
or  =  - — - 

n 

in  the  system  of  notation  introduced  by  Gauss. 

The  principle  may  be  stated  as  follows  :  If  ive  have  n  obsemed 
values  of  an  nnknozvn,  all  equally  good  so  far  as  ive  know,  the 
most  plausible  value  of  tJie  unknoivn  {best  value  on  the  zvhole)  is 
the  arithmetic  mean  of  the  observed  values. 


THE    LAW    OF    ERROR  ii 

It  may  happen  that  the  values  AI^,  M.,,  .  .  .  M„  are  of 
such  nature  that  some  other  symmetrical  function  than  the 
arithmetic  mean  will  satisfy  the  observation  equations  better 
than  will  the  arithmetic  mean.  That  the  arithmetic  mean  is 
on  the  zvhole  the  best  form  of  the  function  may  be  confirmed  by 
a  comparison  of  results  following  from  this  hypothesis  with  the 
records  of  experience. 

7.  By  adding  equations  (i),  Art.  6,  and  taking  the  mean,  wc 
have, 

.  =  M  +  W  =  .,  +  M. 

n  n  n 

The  last  term  of  this  equation  will  become  very  small  if,  ;/  being 
very  large,  the  sum  [A]  of  the  errors  remains  small.  Now,  if, 
after  making  one  observation  and  before  making  another,  we  re- 
adjust our  instrument,  determine  anew  its  corrections,  choose  the 
most  favorable  conditions  for  observing,  and  vary  the  form  of 
procedure  as  much  as  possible,  it  is  reasonable  to  suppose  that 
the  disturbing  influences  will  balance  one  another  in  the  result 
following  from  the  proper  combination  of  the  observed  values. 
It  may  take  an  infinite  number  of  trials  to  bring  this  about.  In 
the  absence  of  all  knowledge,  we  cannot  say  that  it  will  take 
less.  And,  reckoning  blunders  in  reading  the  instrument  or  in 
recording  the  readings  as  accidental  errors,  an  infinity  of  a  higher 
order  than  the  first  may  be  required  to  eliminate  them. 

In  other  words,  there  being  no  reason  to  suppose  that  an  error 
in  excess  (or  positive  error)  is  more  likely  to  occur  on  the  whole 
than  an  error  in  defect  (or  negative  error),  we  may,  when  //  is  a 

very  large  number,  consider  ^  J-  to  be  an  infinitesimal  with  re- 
spect  to  X.     We  may,  therefore,  in  this  case  put 

that  is,  zvhe^t  the  number  of  observed  values  is  very  great,  the 
arithmetic  mean  is  the  true  value. 


12  THE    ADJUSTMENT    OF    OBSERVATIONS 

8.  From  the  principle  of  the  arithmetic  mean,  two  important 
inferences  may  be  derived.  For,  takin^^  the  arithmetic  mean, 
.i-y,  of  n  observed  vakies  of  an  unknown  as  the  most  plausible 
value  of  that  unknown,  we  may  write  our  obsej'vation  equations 
in  the  form, 

x^  —  M.  =  v.,  ,. 


Xo  —  M^  =  i\  J 
where   v^,   v,„  .  .  .  v„   are  called  the  residual  errors  of    obser- 
vation, or  simply  the  residuals, 
{a)    By  addition, 

wxo  -  \M\  =  \v\ 
and  .-. 

M  =  o;  (2) 

that  is,  the  sum  of  the  residuals  is  .zero ;  in  other  words,  the 
sum  of  the  positive  residuals  is  equal  to  the  sum  of  the  negative 
residuals. 

There  is  a  very  marked  correspondence  between  the  series  in 
which  n  is  infinitely  great  and  x  is  the  true  value,  and  a  series 
in  which  ;/  is  finite  and  the  arithmetic  mean  x,,  is  taken  as 
the  best  value  attainable.  For  in  the  first  case  the  sum  of 
the  errors,  A,  is  zero,  and  in  the  second  the  sum  of  the  residuals, 
V,  is  zero. 

{b)  Let  X  be  any  assumed  value  of  the  unknown  other  than 
the  arithmetic  mean,  and  put 

X  -  M,  =  V,'  ^ 
X-  M.  =  v./  ^ 


(3) 


X  -  If „  =  iV 

From  equations  (i)  and  (3),  by  squaring  and  adding, 

[v-]  =  nx,Xo  -  2  x,[M]  +  [M'l 
[(vy]=  nXX  -  2  X  [M]  +  [M'l 

Hence  by  a  simple  reduction, 


THE    LAW    OF    ERROR  13 

[{vr-]  =  [z^]  +  n(^X-^-^J. 
Now,  iX—  — — ^j ,  being  a  complete  square,  is  always  positive. 

••■  [(0^]>M; 

that  is,  t/ic  Sinn  of  tJie  squares  of  the  residuals  v,  foiimi  by  tak- 
ing the  aritJunctic  mean,  is  a  minimum. 

Hence  the  name  Ulethod  of  Least  Squares,  which  was  first 
given  by  Legendre. 

The  Laiu  of  Error  of  Observed  Quantities. 

9.  When  several  independent  measures  of  the  same  quantity, 
all  equally  good,  have  been  made,  it  must  be  granted  that  errors 
in  excess  and  errors  in  defect  are  equally  likely  to  occur  to  the 
same  amount  —  that  is,  are  equally  probable.  Experience  shows 
that  in  any  well-made  series  of  observations,  small  errors  are 
likely  to  occur  more  frequently  than  large  ones,  and  that  there 
is  a  limit  to  the  magnitude  of  the  error  to  be  expected.  If, 
therefore,  a  denotes  this  limit  or  maximum  error,  we  must  con- 
sider all  the  errors  of  the  series  to  be  ranged  between  -\-  a  and 
—  a.  but  to  be  most  numerous  in  the  neighborhood  of  zero. 
Hence  the  probability  of  the  occurrence  of  an  error  may  be  as- 
sumed to  be  a  certain  function  of  the  error. 

If  the  probability  that  an  error  lies  between  o  and  A  be 
denoted  by  /(A),  the  probability  q  of  an  error  between  A  and 
A  +  ^A  is  given  by 

q  =/(;A  -f  d^)  -/(A)  =/'(A)  d^  =  <i>  (A)  dX  suppose  ;        (i) 

q  may  be  taken  to  be  the  probability  of  the  occurrence  of  the 
error  A,  since  dUs.  is  small. 

The  function  (^  (A)  is  called  the  law  of  distribution  of  error, 
or  simply  the  law  of  error. 

The  probability  that  an  error  falls  between  any  assigned 
limits  b  and  a  is  the  sum  of  the  probabilities  ^  (A)  <:/A  extend- 


14        THE  ADJUSTMENT  OF  OBSERVATIONS 

ing  from  b  to  a,  and  is  expressed  in  the  ordinary  notation  of 
the  integral  calculus  by 

<^  (A)  (/A.  (2) 


U  a 


Hence  it  follows  that  the  probability  that  an  error  does  not  ex- 
ceed the  value  a  is 

''^%(A)JA.  (3) 


X" 


The  form  of  <^  (A)  may  be  determined  by  the  aid  of  the  prin- 
ciples of  probability.  For  the  probability  of  the  occurrence  of 
the  error  A,  is  </>  (A,)  ^/A^,  of  A^  is  <^  (A^)  ^A,  .  .  .  .  The 
probability,  Q,  of  the  simultaneous  occurrence  of  the  complete 
system  of  errors  is  the  product  of  the  respective  probabilities  or 

Q  =  «/,(AO<^(A,)  .  .   .   <^(A„)  ,/Aj,  ^A3  .  .   .   ^A„.  (4) 

If  we  make  Q  a  maximum,  we  shall  find  the  most  probable 
value  of  the  unknown  x.  Now  when  g  is  a  maximum,  log  Q  is 
also  a  maximum.  Hence,  differentiating  with  respect  to  Xy  and 
noting  that  the  last  terms  ^A^,  ^A^  .  .  .  <a^A„  do  not  depend 
upon  Xy  we  have, 

^  ^(logQ)  ^  <^^(^)   ^       ^'  (A,)   ^  <^^(A.)   ^ 

^  dx  <^  (A^)    c/x  <^  A3       dv  <^  (A„)    dx 

_    <^'(A.)  <^-(A3)   ^  .   ^:i^  A  (0 


A,<^(Aj)     1       A^c^CA^)     ^  A„</>(A„) 

since  from  Eq.  i,  Art.  6, 

^  _  ^2  _  _  dK  ^ 

dx        dx  dx 

But  from  the  principle  of  the  arithmetic  mean,  when  the 
number  of  observed  values  is  very  great, 

A^  +  A,  -f  .  .  .  +  A,.  =  o.  (6) 

Also,  since  equations  (5)  and  (6)  must  be  simultaneously 
satisfied  by  the  same  value  of  the  unknown,  we  necessarily 
have, 


THE    LAW    OF    ERROR  15 

<f>'(\)     _    <^'(A,)    _  _    <^'(A„) 


=  k,  suppose. 


Hence  for  any  arbitrary  value  A, 

A«^(A)       '• 

Clearing  of  fractions  and  integrating, 
<^  (A)  =  fgi^^-, 

where  e  is  the  base  of  the  system  of  natural  logarithms  and  c  is 
a  constant. 

Again,  since  Q  is  to  be  a  maximum,  -—^  or  — ^^ — -   must 
^  dx^  dx 

be  negative.     Now, 

dx 
d?{\ogQ) 


dx" 


Q  =  c"e-  d\  (/A,. 

Z'(A^  +  A.+   .   .   .  ) 


Hence,  since  n  is  positive,  k  must  be  negative,  and  putting  ^k 
=  —  /r,  we  have 

«/>(A)  =  fe-'^^^^ 

the  law  of  error  sought. 

10.  In  this  expression  there  are  two  symbols  undetermined, 
c  and  /i.  To  find  c.  Since  it  is  certain  that  all  of  the  errors 
lie  between  the  maximum  errors  +  a  and  —  a,  we  have 


X+a 
a 


-/'-AVA=  I. 


But  as  the  values  of  a  are  different  for  different  kinds  of  obser- 
vations, and  as  we  cannot  in  general  assign  these  values  defi- 
nitely, we  must  take  -f  co  and  —  00  as  the  extreme  limits  of 
error,  so  that  c  is  found  from 


X+00 
e-"^-^VA=  I, 
■00 


1 6  THE    ADJUSTMENT    OF    OBSERVATIONS 

and  hence  c  —  — = , 

and  the  law  of  error  may  be  written, 

<^(A)=  ^e-^'^' 
Vtt 

As  regards  h,  it  is  evident  that  for  e"^'^^'^  to  be  a  possible 
quantity,  //  must  be  an  abstract  number.  Hence,  -  is  a  quan- 
tity expressed  in  the  same  unit  of  measure  as  x. 

Also  from  the  form  of  the  function  </>  (A)  it  is  evident  that 
the  probability  of  an  error  A  will  be  the  larger,  the  smaller  //  is, 
and  vice  versa.  Hence,  h  is  a  test  of  the  quality  of  observations 
of  different  series.  It  was  named  by  Gauss  tJic  measure  of 
precision. 

In  practice,  it  is  more  convenient  to  compare  the  precision  of 
different  series  of  observations  by  other  methods. 

II.  Proof  of  the  Law  of  Error  on  Hagen's  (Young's) 
Hypothesis.  —  Various  proofs  of  the  law  of  error  have  been 
derived.  Each  is  open  to  some  theoretical  objection.  The 
following  proof  of  Hagen's  hypothesis  starts  with  a  clear  and 
definite  statement  of  the  assumed  nature  of  an  accidental  error, 
namely,  that  it  is  the  algebraic  sum  of  an  infinite  number  of 
independent  infinitesimal  element  errors,  each  of  which  is  as 
likely  to  be  positive  as  negative.  The  law  of  error  being 
derived  directly  from  this  hypothesis,  it  is  clear  that  if  an  ob- 
server wishes  to  put  the  errors  of  observation  into  this  acci- 
dental class,  and,  therefore,  to  make  them  easy  to  eliminate,  he 
must  use  such  instruments  and  methods  as  will  make  the  errors 
conform  as  closely  as  possible  to  this  definition  of  an  accidental 
error.  It  is  because  Hagen's  proof  thus  indicates  clearly  to  the 
observer  the  standard  toward  which  he  should  struggle,  that  it 
is  here  given  in  addition  to  the  complete  and  independent  proof 
in  the  preceding  article. 

An  accidental  error  of  observation   does  not   result  from  a 


THE    LAW    OF    ERROR  17 

single  cause.  Thus,  in  reading  an  angle  with  a  theodolite,  the 
error  in  the  value  found  is  the  result  of  imperfect  adjustment  of 
the  instrument,  of  various  atmospheric  changes,  of  want  of  pre- 
cision in  the  observer's  method  of  handUng  the  instrument,  etc. 
Each  of  these  influences  may  be  taken  as  the  result  of  numer- 
ous other  influences.  Thus,  the  first  mentioned  may  include 
errors  of  collimation,  of  level,  etc.  Each  of  these  in  turn  may 
be  taken  as  resulting  from  other  influences,  and  so  on.  The 
final  influences,  or  element  errors,  as  they  may  be  called,  must 
be  assumed  to  be  independent  of  one  another,  and  each  as 
likely  to  make  the  resultant  error  too  large  as  too  small  —  that 
is,  as  likely  to  be  positive  as  negative.  The  number  of  these 
element  errors  being  very  great,  we  may,  from  the  impossibility 
of  assigning  the  limit,  consider  it  as  infinite  in  any  case.  Each 
element  error  must  consequently  be  an  infinitesimal,  and  for 
greater  simplicity  we  may  take  those  occurring  in  any  one 
series  as  of  the  same  numerical  magnitude.  Hence  we  con- 
clude that  an  error  of  observation  may  be  assumed  to  be  the 
algebraic  sum  of  a  very  great  number  of  independent  infinitesi- 
mal element  errors  e,  all  equal  in  magnitude,  but  as  likely  to  be 
positive  as  negative. 

Let  the  number  of  these  element  errors  be  denoted  by  2  n, 
as  the  generality  of  the  demonstration  will  not  be  affected  by 
supposing  this  infinitely  great  number  to  be  even.  If  all  of 
the  element  errors  are  +,  the  error  2  //e  results,  and  this  can 
occur  in  but  one  way  ;  if  all  but  one  are  +,  the  error  (2  n  —  2)  e 
results,  and  this  can  occur  in  2  //  ways ;  and  generally,  if 
n  -\-  in  are  +,  and  //  —  ;//  are  — ,   the  error   2  me  results,  and 

^1  •  2  n  (2  n  —  \)  ■  ■  ■  {n  -\-  in  -{-  i)  ^     __ 

this  can  occur  m  ^^ ^ ways.*     Hence 

1 2  •  •  ■  {n  —  in) 

the    numbers    expressing   the  relative  frequency   of  the   errors 

(that  is,  the  number  of  times  they  may  be  expected  to  occur) 

are  equal  to  the  coeflicients  in  the  development  of  the  2  ;/th 

power  of  any  binomial. 

*  See  'rodluintcr's  or  Nfwcomb's   Algebra. 


i8  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  element  errors,  infinite  in  number,  being  infinitely  small 
in  comparison  with  the  actual  errors  of  observation,  these  latter 
may  consequently  be  assumed  to  be  continuous  from  o  to  2  ne, 
the  maximum  error.  If,  therefore,  A  denotes  the  error  in  which 
n  +  in  +  e's  and  n  —  m  —  e's  occur,  and  A  +  ^-/A  denotes 
the  consecutive  error  in  which  u  -^  m  +  1  -f-  e's  and  u  —  vi  —  i 
—  e's  occur,  we  have 

A  =  2  nif. 
A  +  (/A  =  (2  m  +  2)  c, 

and  therefore, 

A  =  wf/A. 

Calling  /  the  relative  frequency  of  the  error  A,  and  /  +  df 
that  of  the  consecutive  error  A  +  ^/A,  we  have 

.      2  w  (2  w  —  i)  •  •  •  (w  +  m  +  i) 

/= ^^ -^^ -t  • 

n  —  m  I 

^  ,     ij-       -.2  w  (2  w  —  i)  •  •  •  (#  +  w  +  2) 


Hence, 

by 

division 

> 

f+df 
f 

or 

df 
f 

n 

—  m 

n 

+ 

2  m  +  I 

n 

+  W  +   I 

2  A  +  (/A 

ndA  +  A  -I-  (/A 
Now,  since  rfA  is  infinitely  small  in  comparison  with  A,  we  may 

write 

d/^  2  A 

/  nd^  +  A  ' 

Also,  since  df  is  infinitely  small  in  comparison  with  /,  2  A  is 
with  respect  to  ndA  -\-  A,  and  we  may  neglect  A  in  the  denom- 
inator in  comparison  with  udA.     We  have,  therefore, 

df^       2  A 


THE    LAW    OF    ERROR 


19 


And  since  A  is  infinitely  small  in  comparison  with  ndX  and 
^/A  is  infinitely  small  in  comparison  with  A,  it  follows  that  Ji 
must  be  an  infinity  of  the  second  order.     It   is,  therefore,  of  a 

magnitude  comparable  with  ., ,  and  hence,  ;/  (^/A)-  must  be 

(«A)' 

a  finite  constant.     Calling  this  constant  — ,  we  have 

/r 

^  =  -  2  h-^d^. 
Integrating  and  denoting  the  value  of  f,  when  A  =  o,  by  /" , 

The  errors  being  separated  by  the  intervals  d\  so  that  o,  d\ 
.  .  .  A,  A  +  rt^A  .  .  .  are  the  errors  in  order  of  magnitude, 
we  must,  in  order  to  make  the  system  consistent  with  the  defi- 
nition of  probability,  and  therefore  continuous,  consider  not  so 
much  the  relative  frequency  of  the  detached  errors  as  the  rela- 
tive frequency  of  the  errors  within  certain  limits. 

Now,  by  the  definition  of  probability,  the  probability  of  an 
error  between  the  limits  A  and  A  -f  rt'A  is  represented  by  a 
fraction  whose  numerator  is  the  number  of  errors  which  fall 
between  A  and  A  +  ^/A,  and  denominator  the  total  number  of 
errors  committed.  If  we  denote  this  probability  by  <^  (A)  we 
may  write 

where  ^  is  a  constant,  2/  being  necessarily  a  constant  for  the 
same  series  of  observations. 

12.  The  Principle  of  Least  Squares.  —  Let  us  return  to  9, 
Kq.  4. 

If  the  observed  values  are  of  the  same  quality  throughout,  // 
is  constant  and  the  product  becomes  c^e~h'^^^'^^.  This  product  is 
evidently  a  maximum  when  [A*]  is  a  minimum  ;  that  is,  if  zvc 
assume  that  cacJi  of  a  very  large  nnnibcr  of  observed  values  of  a 


20  THE    ADJUSTMENT    OF    OBSERVATIONS 

quantity  is  of  the  same  quality,  the  7nost  probable  value  of  the 
quantity  is  found  by  making  the  sum  of  the  squares  of  the  errors 
a  minimum. 

If  the  observed  values  are  not  of  the  same  quahty,  h  is  differ- 
ent for  the  different  observations,  and  the  most  probable  value 
of  the  unknown  would  be  found  from  the  maximum  value  of 
^-[7i2A2].  ^j-^^i-  is,  from  the  minimum  value  of  [/rA''^].  Thus,  if 
each  of  a  large  number  of  observed  values  of  a  quaiitity  is  of 
different  quality,  the  most  probable  value  of  the  quantity  is  found 
by  multiplying  each  error  of  observation  by  its  h,  and  making 
the  sum  of  the  squares  of  the  products  a  mijtimum. 

The  Law  of  Error  of  a  Linear  Function  of  Independently 
Observed  Quantities. 

13.  We  have  found  the  law  of  error  in  the 'case  of  a  quantity 
directly  observed,  and  which  may  be  a  function  of  one  or  more 
unknowns.  There  remains  the  question  as  to  the  form  the  law 
of  error  assumes  in  the  case  of  a  quantity,  F,  which  is  a  linear 
function  of  several  independently  observed  quantities,  M^,  VJ/^, 
.  .  .  M^ ;  that  is,  when 

F  =  ajl,  +  ajl^  +  •  •  •  +  anM„, 

where  a^,  a.„  .  .  .  are  all  constants. 

For  simplicity  in  writing,  consider  two  observed  quantities, 
J/j,  Af,  only,  and  let  h^,  //„  be  their  measures  of  precision. 
The  probability  of  the  simultaneous  occurrence  of  the  errors  A^ 


in  y]/j  and  A^  in  3/^  is 


hlh  e-/'.-V-''rA2=^A,(/A,. 


(i) 


Now,  an  error  Aj  in  M^  and  an  error  A^  in  3f^  produce  an  error 
in  F,  according  to  the  relation 

A  =  OjAj  +  fljA,,  (2) 

and  this  relation  can  always  be  satisfied  by  combining  any  value 


THE    LAW    OF    ERROR  21 

of  Aj  with  all  values  of  A^  ranging  from  —  oo   to  +  oo.      The 
probability,  therefore,  of  an  error  A  in  F  may  be  written, 

But  from  (2),  and  since  A^  is  independent  of  A^, 

(/A  =  a^^A,. 
Hence, 


which  is  of  the  form 


e    hi-a^i  +  h^a^i     (/A, 


Vtt 


That  is,  f/ie  la^v  of  error  of  tJtc  function  F  is  the  same  as  that  of 
the  independently  measured  quantities  7l/, ,  M.,. 
The  precision  of  the  function  F  is  found  from 

}{i  = 1^1112 ; 

that  is,  from 

1  =  ^'  j_  !^  =  F-l 

This  theorem  is  one  of  the  most  important  in  the  method  of 
least  squares,  and  will  be  often  referred  to. 

Ex. — To  find  the   precision   of   the  arithmetic  mean   of   ;/ 
equally  well-observed  values  of  a  quantity  : 

We  have,         F  =  '  {\L  +  M.,+  ■  ■  ■  +.U„). 
n 

Let  //„  =  precision  of  the  arithmetic  mean  ; 

h    =  precision  of  each  observed  value. 


22  THE    ADJUSTMENT    OP    OBSERVATIONS 

or  ho  =  V«/'. 

That  is,  t/ie  precision  of  the  arithmetic  mean  of  n  observations  is 
"sfJi  times  that  of  a  single  observation. 

14.  Comparison  of  the  Accuracy  of  Different  Series  of 
Observations.  —  We  have  seen  that  the  measure  of  precision  h 
affords  a  test  of  the  relative  accuracy  of  different  series  of  ob- 
servations. This  test  vi^as  suggested  by  the  form  of  the  law  of 
error,  and  is  naturally  the  first  that  would  be  chosen  for  that 
purpose. 

The  Mean-Square  Error.  —  On  account  of  the  inconven- 
ience of  computing  h,  Gauss  suggested  the  mean-square  or  "  to 
be  feared  "  error  as  a  test  of  quality.  This  is  defined  as  a 
quantity  /x  whose  square  is  equal  to  the  mean  or  average  of  the 
squares  of  the  individual  errors,  or  when  n  is  very  large. 

^  _  (A2  +  A3^  +  •  •  •  +  A,,^)  _  [A^] 
*^  n  n 

To  find  the  relation  between  //  and  /x.  The  probability  of  the 
occurrence  of  an  error  A,  that  is,  of  an  error  between  A  and  A  + 
rt'A,  is  ^  (A)  di^.  The  number  of  errors  in  the  series  being ;/,  the 
sum  of  the  squares  of  the  errors  in  the  same  interval  will  be 
«A"^  (A)  c/A,  and  the  sum  of  the  squares  of  the  errors  between 
the  limits  of  error  +  a  and  —  a  will  be  for  a  continuous  series. 


A2<^  (A)  JA. 

a 


Extending  the  limits  of  error  ±  ^  to  ±  co  ,  «  being  very  large, 
we  have 


15.   The  Probable  Error. — The  most  common  method  in  use 
in  this  country  of  determining  the  relative  precision  of  different 


THE    LAW    OF    ERROR  23 

series  of  observations  is  by  comparing  errors  which  occupy  the 
same  relative  position  in  the  different  series  when  the  errors  are 
arranged  in  order  of  magnitude.  The  errors  which  occupy  the 
middle  places  in  each  series  are,  for  greater  convenience,  the 
ones  chosen. 

Let  the  errors  in  a  series,  arranged  in  order  of  magnitude,  be 

±  2  a,  ■  •  •  ±  r,  •  ■  ■  o, 

each  error  being  written  as  many  times  as  it  occurs ;  then  we 
give  to  that  error  r  which  occupies  the  middle  place,  and  which 
has  as  many  errors  numerically  greater  than  it  as  it  has  errors 
less  than  it,  the  name  oi  probable  error.  If,  therefore,  n  is  the 
total  number  of  errors,  the  number  lying  between  -|-  r  and  —  r 
is  ;^/2,  and  the  number  outside  these  limits  is  also  «/2.  In  other 
words,  the  probability  that  the  error  of  a  single  observation  in 
any  system  will  fall  between  the  limits  +  r  and  —  ;'  is  1/2,  and 
the  probability  that  it  will  fall  outside  these  limits  is  also  1/2. 
We  have,  therefore, 


ylirJ-r                     2 

from  which  to  find  r. 

If  we  put  //A  =  /,  and  the  value  t  =  p  corresponds  to  A  = 

=  r. 

then 

VttJo                              2 

Expanding  the  integral  in  a  series  (see  Art.  22),  we  shall  find 
that  approximately  the  resulting  equation  is  satisfied  by 

p  =  0.47694. 
Now,  since 

Jir  =  p  =  0.47694  and  A/x  V2  =  i, 

it  follows  that 

r  =  0.6745  /* 

2 
=  *"  fji  roughly. 


24  THE    ADJUSTMENT    OF    OBSERVATIONS 

Hence,  to  find  the  probable  error,  we  compute  first  the  mean- 
square  error  and  multiply  it  by  0.6745. 

As  a  check,  the  error  which  occupies  the  middle  place  in  the 
series  of  errors  arranged  in  order  of  magnitude  may  be  found. 
It  will  be  nearly  equal  to  the  computed  value,  if  the  seriesis  of 
considerable  length. 

It  is  to  be  clearly  understood  that  the  term  probable  error 
does  not  mean  that  that  error  is  more  probable  than  any  other, 
but  only  that  in  a  future  observation  the  probability  of  commit- 
ting an  error  greater  than  the  probable  error  is  equal  to  the 
probabihty  of  committing  an  error  less  than  the  probable  error. 
Indeed,  of  any  single  error  the  most  probable  is  zero.  Thus  the 
probability  of  the  error  zero  is  to  that  of  the  probable  error  r  as 

h         h 


e 


/i-r2 


or  I  :  e- (0-47694)2^ 

or  I  :  0.8. 

The  idea  of  probable  error  is  due  to  Bessel  {Berlin.  Astron. 
Ja/irb.,  18 1 8).  The  name  is  not  a  good  one,  on  account  of  the 
word  "probable"  being  used  in  a  sense  altogether  different  from 
its  ordinary  signification.  It  would  be  better  to  use  the  term 
critical  error,  for  example,  as  suggested  by  De  Morgan,  or 
median  error,  as  proposed  by  Cournot. 

16.  The  Average  Error.  —  It  naturally  occurs  as  a  third  test 
of  the  precision  of  different  series  of  observations,  to  take  the 
mean  of  all  the  positive  errors  and  the  mean  of  all  the  negative 
errors,  and  then,  since  in  a  large  number  of  observations  there 
will  be  nearly  the  same  number  of  each  kind,  to  take  the  mean 
of  the  two  results  without  regard  to  sign.  This  gives  what  may. 
be  termed  the  average  error.  It  is  usually  denoted  by  the  Greek 
letter  -q,  so  that 

[A 

where  [A  is  the  arithmetic  sum  of  the  errors. 


THE    LAW    OF    ERROR  25 

An  expression  for  7;  in  terms  of  the  mean-square  error  [x  may 
be  found  as  follows.  The  number  of  errors  between  A  and 
A  +  ^/A  is 

n^  (A)  (/A, 

and  the  sum  of  the  positive  errors  in  the  series  is 


Jr»oo 
A<^  (A)  dA. 
0 


The  sum  of  the  negative  errors  being  the  same,  the  sum  of 
all  the  errors  is 


■£ 


Ac/)  (A)  r/A. 

Hence  v  =  ^  l       -^<^  (^)  ^^ 

2 


•+  =0 


=,^  r  Ac-''=-^vA 

VttJo 


=  7-7-  =  ^  V  - » 

the  relation  required. 

The  average  error  may,  as  stated  above,  be  directly  used  as  a 
test  of  the  relative  accuracy  of  different  series  of  observations. 
The  general  custom  is,  however,  to  employ  it  as  a  stepping-stone 
to  find  the  mean-square  and  probable  errors.  This  can  be  done, 
for  the  reason  that  it  is  more  easy  to  compute  [A  than  [A']. 

17.  The  formulas  for  /x  and  r  computed  in  this  way  are  as 
follows.     From  the  last  equation  preceding 

/*  =  \  2  ^ 

=  1-2533  ^^-' 

and  from  Art.  14,  r  =  0.6745  /x 

=  0.8453  ^;J'- 

The  relations  connecting  /u.,  r,  and  ?/  are  easily  remcmbcied   in 
the  f(j]lowing  form  : 


26 


THE    ADJUSTMENT    OP    OBSERVATIONS 


fjL  V2  =   -  =  Vttj;. 
P 

These  relations  may  also  be  conveniently  arranged  in  tabular 
form  : 


M 

?■ 

V 

fl  = 
r  = 

V  = 

1 .0000 
0.6745 
0.7979 

1.4826 
1 .0000 
I. 1829 

1-2533 
0.8453 

1 .0000 

The  p.  e.  which  is  to  be  used  as  a  measure  of  accuracy  may 
be  computed  from  the  sum  of  the  squares  of  the  errors  and  also 
from  the  sum  of  the  errors  without  regard  to  sign.  The  question 
then  arises,  which  of  the  two  methods  will  give  the  better  re- 
sult }  It  may  be  stated  without  here  setting  forth  the  proof, 
that  the  value  of  the  p.  e.  computed  from  the  squares  is  some- 
what more  trustworthy  than  it  is  when  derived  from  the  first 
powers  of  the  errors. 

18.  Whether  we  should  use  the  m.  s.  e.  or  the  p.  e.  in  stating 
the  precision  is  largely  a  matter  of  taste.  Gauss  says  :  "  The 
so-called  probable  error,  since  it  depends  on  hypothesis,  I,  for  my 
part,  would  like  to  see  altogether  banished  ;  it  may,  however,  be 
computed  from  the  mean  by  multiplying  by  0.6744897."  On 
the  other  hand,  the  International  Committee  of  Weights  and 
Measures  decided  in  favor  of  the  probable  error  :  "  It  has  been 
thought  best  in  this  work  that  the  measure  of  precision  of  the 
values  obtained  should  always  be  referred  to  the  probable  error 
computed  from  Gauss'  formula,  and  not  to  the  mean  error." 
{Proch  Verbanxy  1879,  p.  TJ.) 

In  the  United  States,  in  the  Naval  Observatory,  the  Coast 
Survey,  the  Engineer  Corps,  and  the  principal  observatories,  the 
p.  e.  is  used  altogether.  So,  too,  in  Great  Britain,  in  the  Green- 
wich Observatory,  the  Ordnance   Survey,    etc.     In  the  G.  T. 


THE    LAW    OF    ERROR  27 

Survey  of  India  the  m.  s.  e.  is  used,  for  the  reason  given  by 
Gauss  above.  Among  German  geodeticians  and  astronomers 
the  m.  s.  e.  is  very  generally  employed. 

The  p.  e.  has  a  definite  meaning,  namely,  that  the  chances  are 
even  for  and  against  a  given  error  being  greater  or  less  than  the 
corresponding  p.  e.  This  frequently  furnishes  a  convenient  test 
as  to  whether  the  errors  of  a  given  series  of  observations  are  dis- 
tributed according  to  the  assumed  law  of  error. 

The  p.  e.  will  be  used  in  the  text  of  this  book  nearly  always. 

TJie  Probability  Ciui^e. 

19.  The  principles  laid  down  in  the  preceding  articles  may 
be  illustrated  geometrically  as  follows  : 

We  have  seen  that  in  a  series  of  observations  the  probability 
of  an  error  A,  that  is,  that  an  error  will  lie  between  the  values 
A  and  A  +  <;/A,  is  given  by  the  expression  (Art.  13) 

Vtt 

Now,  if  O  is  the  origin  of  coordinates,  and  a  series  of  errors, 
A,  are  represented  by  the  distances 
from  O  along  the  axis  of  abscissas 
OX,  positive  errors  being  taken  to  the 
right  of  O  and  negative  errors  to  the 
left,  then  the  probability,  in  a  future 
observation,  of  an  error  falling  be- 
tween A  and  A  +  ^/A,  will  be  repre-  Pig  , 

sented  by  the  rectangle  whose  height  is— ^^  <?-/i-'A=   and    width 

Vtt 

rt'A,  or,  more  strictly,  by  the  ratio  of  this  rectangle  to  tlie  sum 

of    all  such  rectangles  between    the    extreme    limits  of    error. 

This  sum  we  have  for  convenience  already  denoted  by  unity. 

Hence,  for  a  series  of  obserx'ations  whose  ciuality  is  known, 

by  giving  to  A  all  values  from  +  co  to  —  00  and    drawing  the 


28  THE    ADJUSTMENT    OF    OBSERVATIONS 

corresponding-  ordinates,  we  shall  have  a  continuous  curve  whose 
equation  may  be  written 

This  curve  is  called  the  probability  curve. 

20.  To  Trace  the  Form  of  the  Curve.  —  Since  A  enters  to 
the  second  power,  and  j  to  the  first  power,  the  curve  is  symmet- 
rical with  respect  to  the  axis  of  j,  and  the  form  of  the  equation 
shows  that  it  lies  altogether  on  one  side  of  the  axis  of  A.     Also, 

when  A  =  o,  —  =  o ;  that  is,  the  tangent  at  the  vertex  is  par- 

allel  to  the  axis  of  x. 

As  A  increases  from  o  the  values  of  y  continually  decrease. 
When  A  =  ±  CO  ,  then 

y  =  o    and   dy/d^  =  o, 
showing  that  the  axis  of  A  is  an  asymptote. 

Again,  since 

—^  =  —7=e    "  -^   (2  //-A''  —  i), 
di^        Vtt 

there  is  a  point  of  inflection  when 

.  I 

A  =  y-y=  =  fl, 
'I  •V2 

and  the  m.  s.  e.  is  therefore  the  abscissa  of  the  point  of  inflection. 
Also,  when  A=:0,  tfy/dA^  is  negative,  showing  that  the  ordinate 
at  the  vertex  is  the  maximum  ordinate.  Hence  the  curve  is  of 
the  form  indicated  in  Fig.  i,  OA  representing  the  maximum 
ordinate  and  Oil/ the  m.  s.  e. 

The  values  of  //,  that  is,  of  ij [1^2,  being  different  for  differ- 
ent series  of  observations,  the  form  of  the  curve  will  change  foi 
each  series,  and  the  curve  may  be  plotted  to  scale  from  values 
of  y  corresponding  to  assumed  values  of  A. 

In  plotting  the  curve,  since  the  maximum  ordinate  at  the  ver- 
tex //  Vvr  enters  as  a  factor  into  the  values  of  each  of  the  other 
ordinates,  its  value  may  be  arbitrarily  assumed.     We  may  there- 


^^.' 


THE    LAW    OF    ERROR 


29 


fore  adopt  a  scale  for  plotting  the  ordinates  different  from  the 
scale  by  which  the  abscissas  are  plotted,  in  order  to  show  the 
curve  more  clearh'. 

The  iovm  of  the  curve  is  in  accordance  with  the  principles 
already  laid  down  in  deducing  the  law  of  error,  and  could  have 
been  derived  from  them  directly.  Thus,  that  small  errors  are 
more  probable  than  large,  is  indicated  by  the  element  rectangle 
areas  being  greater  for  values  of  A  near  zero  than  for  values 
more  distant ;  that  very  large  errors  have  a  very  small  probability 
is  indicated  by  the  asymptotic  form  of  the  curve ;  and  that  posi- 
tive and  negative  errors  are  equally  probable,  is  indicated  by  its 
symmetrical  form  with  respect  to  the  axis  of  j. 

21.    The  area  of  the  curve  of  probability  is  the  sum  of  the 

rectangles — =  e~'^'-^'c/A,  for  values  of  A  extending  from  -(-  00  to 

Vtt 

—  00  ,  and  may  be  denoted  by  unity.  If,  then,  we  represent  by 
their  area  the  total  number  of  errors  that  occur  in  a  series  of 
observations,  it  follows  from  the  definition  of  probability  that 
the  area  included  between  certain  assigned  limits  will  represent 
the  number  of  errors  to  be  expected  in  the  series  between  the 
values  of  those  lim.its. 

Thus,  if  O  is  the  origm,  the  area  to  the  right  of  OA  would 
represent  the  number  of  positive  errors, 
and  the  area  to  the  left  of  OA  the  num- 
ber of  negative  errors.  The  area  OPP' A 
would  represent  the  number  of  positive 
errors  less  than  OP,  the  area  PRR' P' 
the  number  that  lie  between  OP  and 
OR. 

If  the  area  AOPP'  is  equal  to  one-half  the  total  area  AOX, 
then  the  number  of  positive  errors  less  than  OP  would  be  equal 
to  the  number  greater  than  OP.  Hence  OP  would  represent 
the  probable  error.  If  OQ  be  taken  equal  to  OP,  the  area 
PQQ P'  would  represent  the  number  of  errors  numerically  k-ss 
than  the  probable  error. 


Fig.  2. 


30  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  average  error  ?;  is  evidently  represented  by  the  abscissa 
of  the  center  of  gravity  of  either  the  positive  or  negative  half  of 
the  probability  curve. 
The  Law  of  Error  Applied  to  an  Actual  Series  of  Observations. 

We  here  bridge  over  the  gulf  between  the  ideal  series  from 
which  we  have  derived  the  law  of  error,  and  the  actual  series 
with  which  we  have  to  deal  in  practical  work,  and  which  can 
only  be  expected  to  come  partially  within  the  range  of  the  law 
constructed  for  the  ideal. 

22.   Effect  of  Extending  the  Limits  of   Error  to  ±  go. — 

The  expression  -^  ^-'''^VA  gives  the  value  of  the  probability 

of  an  error  between  A  and  A  4-  <:/A  in  an  ideal  series  of  observa- 
tions where  the  values  are  continuous  between  limits  infinitely 
great.  In  all  actual  series  the  possible  error  is  included  within 
certain  finite  limits,  and  the  probability  of  the  occurrence  of  an 
error  beyond  those  limits  is  zero.  Practically,  however,  the  ex- 
tension of  the  limits  of  error  to  ±  co  can  make  no  appreciable 
difference  in  either  case,  as  the  function  </)  (A)  decreases  so 
rapidly  that  we  can  regard  it  as  infinitesimal  for  large  values  of 
A ;  in  other  words,  the  greater  number  of  errors  is  in  the  neigh- 
borhood of  zero,  and  therefore  the  most  important  part  is  the 
part  covered  by  both.  This  has  been  illustrated  geometrically 
in  the  discussion  of  the  probability  curve,  and  will  now  be  de- 
veloped from  another  point  of  view. 

The  probability  of  the  occurrence  of  an  error  not  greater 
than  rt  in  a  series  of  observations  is,  since  the  error  must  lie 
between  +  a  and  —  a, 


—^   \        e-'''^'d^  (see  Art.  13). 


This  is  the  same  as  the  area  of  the  curve  of  probability  be- 
tween the  limits  -^a  and  —a,  the  total  area  being  unity  (Art.  13). 
Change  the  variable  by  placing  /-A  =  t.  The  expression  then 
becomes, 


THE    LAW    OF    ERROR  31 

^       f  ha  2        /* ''« 

—^  I      e-^'dt    or    -~   I      e-^'dt, 

and  is  usually  denoted  by  the  symbol  0  (/). 

If,  as  in  Art.  14,  the  value  t  =  p  corresponds  to  A  =  r,  we  have 
finally,  by  eUminating  h, 


© 


Jr.P- 
r 


The  value  of  this  integral  cannot  be  expressed  exactly  in  a 
finite  form,  but  may  be  found  approximately  as  follows  : 

Expanding  e~^'  in  a  series,  and  integrating  each  term  sepa- 
rately, ^ve  have, 

f'e-''dt=   rYi--  +  — ]dt 

Jo  Jo    V  I  1-2  / 

3       1-2    5 

This  series  is  convergent  for  all  values  of  /,  but  the  conver- 
gence is  only  rapid  enough  for  small  values  of  A 

For  large  values  of  t  it  is  better  to  proceed  as  follows : 

Integrating  by  parts, 

fe-''dt=    C-~dc-'' 
J  J       21 

= e~^^ /  —^rdi 

2  t  2jt- 

2  t  2^f  2V       r 

Hence    f  V'V/  =  ^' S  i  -  ^  +  -1^1-,  -  ^}  +  .  .  .\ 

J  a  2a   I  2  I-         (2  f)-         (3  l-y  ^ 

But  /    e-^'^dt  =    I  e-^'dt  -    /     er'^dt 

Jo  Jo  Jt 

2    Jt 


-^'di 
2    Jt 

•.   finally, 


i: 


-12  „       Vtt       e   "^  I  1.3  I-3-.S    ,  } 

'  '^'==^-Ti('-z?^(2Pr~{^ir^"'S 


32  THE    ADJUSTMENT    OF    OBSERVATIONS 

23.    Approximate  values  of  the  expression  /    ^~'W/  may  be 

computed  from  the  above  formulas  for  any  numerical  value  of  t. 
In  Table  I  (Art.  213)  will  be  found  the  values  of  the  func- 
tion 0  (/)  corresponding  to  the  argument  a  jr.     The  reason  for 
arranging  the  table  in  this  way  is  that  it  is  more  convenient  to 

compute  -  than  o     ,  where 
r  r 

p  —  0.47696. 

The  probability  that  an  error  exceeds  a  certain  error  a  is 
I  _  0  (/),  and  may  be  found  from  Table  I  by  deducting  the 
tabular  value  from  unity.  Thus  we  have  the  probability  that  a 
is  greater  than  r  is  0.5,  than  2  r  is  0.177,  than  3  r  is  0.043,  than 
4  r  is  0.007,  than  5  r  is  o.ooi,  than  6  r  is  0.000 1. 

Hence,  in  10,000  observations  we  should  expect  only  one 
error  greater  than  6  /-,  in  1000  only  one  greater  than  5  r,  in  100 
only  one  greater  than  4;',  and  in  25  only  one  greater  than  3  r. 
If  in  any  set  of  observations  we  found  results  much  at  variance 
with  these,  we  could  assume  that  they  arose  from  some  unusual 
cause,  and  should,  therefore,  be  specially  examined.  As  in 
practice  the  number  of  observations  in  any  case  is  usually  under 
100,  we  are  eminently  safe  in  taking  the  maximum  error  at 
about  5  ;-  or  3  /x. 

Experience  indicates  that  in  general  the  curve  representing 
the  true  law  of  error  for  a  given  series  of  observations  departs 
but  little  from  the  Gaussian  curve.  The  actual  curve  for  a  given 
series  may  quickly  be  compared  with  the  Gaussian  curve  by  the 
use  of  Table  I.  The  degree  of  departure  from  the  Gaussian 
curve  necessarily  indicates  the  extent  to  which  the  facts  differ 
from  the  assumption  of  an  infinite  number  of  sources  of  infini- 
tesimal errors. 

If  a  set  of  observations  shows  a  marked  divergence  from  this 
law,  a  rigid  examination  will  reveal  the  necessity,  in  general,  of 
applying  some  hitherto  unknown  correction.  Thus,  in  the  earl- 
ier differential  comparisons  of  the  compensating  base-apparatus 


THE    LAW    OF    ERROR  33 

of  the  United  States  Lake  Survey  with  the  standard  bar  packed 
in  ice,  the  observed  differences  did  not  follow  the  law  of  error,  as 
it  was  fair  to  suppose  that  they  should,  the  bars  being  compen- 
sating. There  was  instead  a  regular  daily  cycle  ;  some  one 
source  of  error  so  far  exceeded  the  others  that  it  overshadowed 
them.  A  study  of  the  results  was  made,  and  the  law  of  daily 
change  disco\-ered,  which  gave  a  means  of  applying  a  furthtr 
correction.  The  work  done  later,  after  taking  account  of  this 
new  correction,  showed  nothing  unusual. 

24.    General  Conclusion.  —  On  the  whole,  though  we  cannot 

say  that  the  formula  -^_  i'-'''^'  will  truly  represent  the  law  of 

error  in  any  given  series  of  observations,  we  can  say  that  it  is  a 
close  approximation. 

When  in  a  series  of  observations  we  have  exhausted  all  of  our 
resources  in  finding  the  corrections,  and  have  applied  them  to 
the  measured  values,  the  residuum  of  error  may  fairly  be  sup- 
posed to  have  arisen  from  many  sources  ;  and  we  conclude  from 
the  foregoing  investigations  that,  of  any  one  single  law,  the  best 
to  which  we  can  consider  the  residual  errors  subject,  and  the 
best  to  be  applied  to  a  set  of  observations  not  yet  made,  is  the 
exponential  law  of  error. 

The  general  theorem  of  Art.  12  may  therefore  be  applied  to 
a  limited  series  and  be  written  :  If  tJic  observed  values  of  a  quan- 
tity are  of  different  quality,  tJie  most  probable  value  is  found  by 
dividing  each  residual  error  by  t lie  probable  error  and  making  the 
sum  of  the  squares  of  the  quotients  a  miniinum  ;  if  of  the  same 
quality,  the  most  probable  value  is  the  arithmetic  mean  of  the  ob- 
served values. 

Classifieation  of  Observations 

25.  For  purposes  of  reduction,  observations  may  be  divided 
into  two  classes  —  those  which  are  independent,  being  subject  to 
no  conditions  except  those  fixed  by  the  observations  themselves; 
and  those  which  are  subject  to  certain  conditions  out.side  of  the 


34  THE    ADJUSTMENT    OF    OBSERVATIONS 

observations,  as  well  as  to  the  conditions  fixed  by  the  observa- 
tions. In  the  former  class,  before  the  observations  are  made, 
any  one  assumed  set  of  values  is  as  likely  as  any  other  ;  in  the 
latter  no  set  of  values  can  be  assumed  to  satisfy  approximately 
the  observation  equations  which  does  not  exactly  satisfy  the 
a  priori  conditions. 

For  example,  suppose  that  at  a  station  O  the  angles  AOB, 
A  OC,  are  measured.  If  the  measures  of  each  angle  are  inde- 
pendent of  those  of  the  other,  the  angles  are  found  directly. 

The  angle  B OC  co\x\d  be  determined  from  the  relation 

AOC=AOB  +  £0C. 

The  unknown  in  this  case  may  be  said  to  be  observed  indirectly, 
and  therefore  independent  observations  may  be  classed  as  direct 
and  indirect.     The  former  class  is  a  special  class  of  the  latter. 

But  if  the  angle  BOC  is  observed  directly  as  well  as  AOB, 
AOC,  then  these  angles  are  no  longer  independent,  but  are  sub- 
ject to  the  condition  that  when  adjusted 

AOC ^  AOB  +  BOC, 

and  no  set  of  values  can  be  assumed  as  possible  which  does  not 
exactly  satisfy  this  condition. 

The  observations  in  this  case  are  said  to  be  conditioned. 
Though  we  have,  therefore,  strictly  speaking,  only  two  classes 
of  observations,  we  shall,  for  simplicity,  divide  the  first  into  two, 
and  consider  in  order  the  adjustment  of 

(i)    Direct  observations  of  one  unknown. 

(2)  Indirect  observations  of  several  independent  unknowns. 

(3)  Conditioned  observations. 


CHAPTER    III 

ADJUSTMENT    OF    DIRECT    OBSERVATIONS    OF    ONE 
UNKNOWN    QUANTITY 

In  the  application  of  the  ideal  formulas  of  Chapter  II  to  an 
actual  series  of  observ^ations,  we  shall  begin  with  a  single  quan- 
tity which  has  been  directly  observed.  We  shall  consider  two 
cases  —  first,  when  all  of  the  observed  values  are  of  equal  qual- 
ity, and,  next,  when  they  are  not  all  of  equal  quality. 

There  are  in  all  cases  two  quantities  to  be  found  —  first  the 
most  probable  value  of  the  unknown  itself,  and  next  the  pre- 
cision of  this  value. 

A.    Observed   Vahies  of  Equal  Quality. 

26.    The  Most  Probable  Value;  the  Arithmetic  Mean. — V/e 

have  seen  that  in  a  series  of  directly  observed  values  J/,,  J/,, 
.  .  .  J/„,  of  equal  quality,  the  most  probable  value  x^^  of  the 
observed  quantity  is  found  by  taking  the  arithmetic  mean  of 
these  values  ;  that  is, 

Xo=[M]/«.  (i) 

It  has  also  been  shown  that  the  same  result  will  follow  by 
making  the  sum  of  the  squares  of  the  residual  errors  a  mini- 
mum.    (Art.  12.) 

As  the  observed  values  vJf  are  often  numerically  large  and  not 
widely  different,  the  arithmetical  work  of  finding  the  mean  may 
be  shortened  as  follows  : 

A  cursory  examination  of  the  observations  will  show  about 
what  the  mean  value  .x\^  must  be.  Let  X'  denote  this  approxi- 
mate value  of  .i',|,  which  may  conveniently  be  taken  some  round 
number.     Subtract  each  of  the  observed  values  J/,,  I\f.„  .  .  .  J/„ 

35 


36  .   THE    ADJUSTMENT    OF    OBSERVATIONS 

in  succession  from  X'  and  call  the  differences  Z^,  l,„  .  .  .  4 
respectively.     Then, 

X'-M,  =  l„ 

X'  -  AU_  =  /,,  (2) 

X'  -  M„  =  /„. 
By  addition,  nX'  -  [M]  =  [/]. 

But  .o  =  W. 

n 

11 
=  A''+  .v',  suppose.  (3) 

Hence  all  that  we  have  to  do  is  to  take  the  mean  x'  of  the  small 
quantities,  /;,  4,  .  •  .  4>  ^^'^^l  add  the  assumed  value  X'  to  the 
result. 

27.  Control  of  the  Arithmetic  Mean.  —  In  least-square  com- 
putations, it  is  important  to  have  a  check  or  control  of  the 
numerical  work.  This  is  specially  desirable  when  a  computation 
takes  several  weeks,  or  it  may  be  months,  to  complete  it.  In 
long  computations  it  is  better  for  two  computers  to  work  together, 
using  different  methods  whenever  possible,  and  to  compare 
results  at  intervals.  But  even  this  is  not  an  absolute  safeguard 
against  mistakes,  as  it  sometimes  happens  that  both  make  the 
same  slip,  as,  for  example,  writing  +  for  — ,  or  vice  versa.  Hence, 
even  if  the  computation  is  made  in  duplicate,  it  is  advisable 
to  carry  through  an  independent  check  which  may  be  referred 
to  occasionally.  In  computations  not  duplicated  a  control  is 
essential. 

A  control  of  the  accuracy  of  the  arithmetic  mean  of  a  set  of 
observed  values  of  the  same  quantity  is  afforded  by  the  relation 

[v]  =  o  ; 

that  is,  that  the  sum  of  the  positive  residuals  should  be  equal  to 
the  sum  of  the  negative  residuals. 

If,  however,  in  finding  the  arithmetic  mean,  the  sum  [J/]  of 
the  observed  quantities  was  not  exactly  divisible  by  their  num- 
ber ;/,  the  sums  of  the  positive  and  negative  residuals  would  not 


OBSERVATIOXS    OF    ONE    UXKXOWX    QUAXTITV         37 

be  equal,  but  the  amount  of  the  discrepancy  could  easily  be 
estimated  and  allowed  for.  For  if  the  value  of  the  mean  taken 
were  too  large  by  a  certain  a-mount,  the  positive  residuals  would 
each  be  too  large,  and  the  negative  residuals  too  small,  by  that 
amount.  Hence  the  discrepancy  to  be  expected  would  be  //  times 
the  amount  that  the  approximate  quotient  taken  as  the  mean 
differed  from  the  exact  quotient. 

Ex.  —  In  the  telegraphic  determination  of  the  difference  of  longitude  be- 
tween St.  Paul  and  Duluth,  Minn.,  June  15,  1S71,  the  following  were  the 
corrections  found  for  chronometer  Bond  No.  176  at  15  h.  51  m.  .sidereal 
time  from  the  observations  of  21  time  stars.     {Report  Chief  of  Engineers, 

U.S  A.,  1S71.) 


M 

V 

z!" 

J- 

s 

-8.78 

+  0.04 

0.0016 

.76 

•f  .02 

4 

•85 

+   .11 

121 

.78 

+    .04 

s 

16 

•51 

-0.23 

5-9 

,64 

—    .10 

100 

.68 

-  .06 

36 

■63 

—    .11 

121 

■58 

-  .16 

256 

.80 

+    .06 

36 

•75 

+       .01 

I 

.78 

+  .04 

16 

.96 

+      .22 

4S4 

.64 

—    .10 

100 

•65 

-  .09 

81 

•83 

+      .09 

81 

.70 

-   .04 

16 

.64 

—  O.IO 

100 

•79 

+    -05 

25 

.90 

+    .16 

256 

-8.93 

+  0.19 

0.0361 

Mean 

-8.74 

+  1.03 
\v  =  2.02 

-0.99 

[7/=]  =  .02756 

Taking  the  observations  as  of  equal  precision,  we  find  the  arithmetic 
mean  to  be  -  8.74.     This  is  the  most  probable  value  of  the  correction. 

The  residuals  7/ are  found  by  subtracting  each  observed  value  from  tiu- 
most  probable  value  according  to  the  relation, 


41fi4.^4 


38  THE    ADJUSTMENT    OF    OBSERVATIONS 

X'  -  M^v. 

They  are  written  in  two  columns  for  convenience  in  applying  the  check, 

[v]  =  o. 

The  true  mean  may  be  derived  by  subtracting  the  mean  of  the  residuals 
from  the  approximate  mean. 

28.  It  was  desired  in  this  case  to  secure  a  mean  which  is 
correct  in  the  second  decimal  place,  or,  in  other  words,  is  not  in 
error  by  more  than  5  in  the  third  decimal  place.  This  has  ob- 
viously been  secured  when  the  difference  between  the  sums  of 
the  +  and  —  residuals  is  less  than  (0.005)  (21)   =  0.105. 

In  general,  a  mean  is  coiTcct  to  a  given  decimal  place  wJien  the 
difference  betzveeji  the  sums  of  the  -f-  and  —  residuals  is  less, 
expressed  in  units  of  that  decimal  place,  than  one-half  of  the 
number  of  quantities  of  ivJiicJi  the  mean  is  sought. 

If  the  difference  of  the  sums  of  the  4-  and  —  residuals  be 
found  to  be  too  great,  an  error  has  been  made  either  in  deriv- 
ing and  adding  residuals,  or  in  deriving  the  mean.  If  it  is 
believed  to  be  the  latter,  or  if  the  value  used  in  deriving  the 
residuals  was  only  an  assumed  approximate  value,  the  true  mean 
may  be  quickly  derived  by  subtracting  the  mean  value  of  the 
residuals  from  the  assumed  mean  used  in  finding  the  residuals. 

29.  Precision  of  the  Arithmetic  Mean.  —  The  degree  of 
confidence  to  be  placed  in  the  most  probable  value  of  the  un- 
known is  shown  by  its  probable  error. 

■  /         {a)    BesseVs  Formula. 

If  we  knew  the  true  value  x  of  the  unknown,  and  conse- 
quently the  true  errors  A^,  A^  .  .  .,  we  should  have,  as  in  Art. 
20,  for  the  m.  s.  e.  of  an  observation, 

n 

But  we  have  only  the  most  probable  value  x^  and  the  residuals 
i\,  v.„  .  .  .  Vn  instead  of  the  true  values  x,  A^,  A^  .  .  .  A^. 
Now, 

Xf^  —  V^  =  If  1  =  X—  \^ 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY        39 
x,-v,  =  M,  =  x-  \,  (i) 


Xo  —  I'n  =  ^V„  =  -V  —  A„. 

By  addition,  remembering  that  [t']  =  o, 

w.Vo  =  nx  -  [A].  (2) 

Substitute  for  x^  from  equation  (2)  in  equations  (i)  and 

W77j  =  (h  —  i)  A^  —  A^  —  •  .  • 

nv^  =       —  Aj  +  (w  —  i)  A,  —  .  •  . 


Squaring, 

«2^^2   ^    („    _    1)2  ^^2    _|.  A/    +  ...    -    2(W  -    l)  AjA, 

n^T;,-  =  ^{  +  ("  -  i)-  A/  +  ...  -  2(»  -  i)  A.A,  -  •  •  • 

By  addition,  assuming  that  the  double  products  destroy  each' 
other,*  positive  and  negative  errors  being  equally  probable, 

n^v-"-]  =  {{n  -  ly  +  (w  -  i)}[A2]. 
...[.2]=^  [A3] 

=  (w  —  I)/A^ 

and  i^'  =  -t3_ ,  (3) 

n  —  I 

which  gives  the  m.  s.  e.  of  an  observation. 
Now,  from  Art.  13,  14, 

/*o  =  — P* 


T    W(W  —   l) 

which  gives  the  m.  s.  e.  of  the  arithmetic  mean  of  ;/  observa- 
tions of  equal  precision. 

*  If  the  positive  and  negative  errors  are  equal  in  number,  «,  tiicre  will 
be  a  preponderance  of  negative  products.  This  is  too  slight  to  affect  the 
proof.  For  the  proportion  of  the  excess  of  the  negative  products  to  the 
total  number  of  products  is  as  n  to  2  «^  -ti,  or  as  i  to  2«  -  i,  which  de- 
creases as  the  number  of  errors  2  ?i  increases. 


46  THE    ADJUSTMENT    OF    OBSERVATIONS 

30.  From  the  constant  relation  existing  between  the  m.  s.  e. 
and  p.  e.  given  in  Art.  14,  we  have  for  the  p.  e.  of  an  observa- 
tion and  of  the  arithmetic  mean  of  n  observations  respectively, 


=  />V^V/::^-  » 


where  p  ^2  =  0.6745  nearly. 

31.  It  is  important  to  note  carefully  the  distinction  between 
residuals  and  errors.  The  errors  are  quantities  of  which  we  may 
never  hope  to  secure  the  exact  values,  since  they  are  the  differ- 
ences between  the  true  value  and  the  separate  observed  values. 
We  cannot  secure  the  true  value.  We  can  secure  a  most  prob- 
able value  which  is  an  approximation  to  the  true  value.  The 
residuals  are  the  differences  between  the  most  probable  value 
and  the  separate  observed  values.  The  residuals  are  approxima- 
tions to  the  corresponding  errors  just  as  the  most  probable  value 
is  an  approximation  to  the  true  value.  The  residuals  are  quan- 
tities which  may  be  used  in  computations.  The  errors  cannot 
be  so  used  since  they  are  always  unknown.  The  errors  appear 
in  the  formulas  during  the  process  of  deriving  them,  but  they 
necessarily  disappear  from  the  formulas  before  they  are  in  shape 
to  be  used  by  the  computer. 

32.  Peters'  Formula.  — The  m.  s.  e.  and  p.  e.  of  a  series  of 
observed  values  may  be  more  rapidly  computed  from  the  sum  of 
the  errors  rather  than  from  the  sum  of  their  squares  by  means 
of  the  convenient  formula  first  given  by  Dr.  Peters.* 

From  the  equation, 

«  —  I 

we  have  approximately,  without  regard  to  sign, 

*  Asiroiio/nisc/ie  A'ac/irickten,  No.  1034. 


OBSERVATIONS    OP    OXE    UXKXOWN    QUAXTITY        41 


''1 

\- 

n 

^\, 

7;, 

i/'^ 

— 

'a 

\ 

n 

2' 

[v 

\/-'^ 



I 

—  V, 

Adding  and  dividing  by  ;/, 


where  [v  is  the  sum  of  the  residuals  without  regard  to  sign. 
But  from  Art.  16, 

H-=\  -V- 


■'■^■^       [v  nearly. 


V«  {n  —  1) 

For  the  p.  e.  of  an  observation  and  of  the  arithmetic  mean  of 
n  observations  we  have  respectively 

y-  0.8453      r 

Vw  {n  —  1) 

0-8453     , 
ro  = F==  \v. 


n  Nn  —  I 


33.  Collecting  the  formulas  for  finding  the  p.  e.  of  a  single 
observation  and  of  the  arithmetic  mean  of  ;/  observations,  we 
have 

r  =  0.6745  v/J^  .  ''  =  °-^453     ,^- — -^  » 

\  n  —  I  yn(n—  I) 

r„  =  0.67451/ \}-} ,  ro  =  0.8453—''^        • 

To  save  labor  in  the  numerical  work,  I  have  computed  tables 
containing  the  values  of  the  coefficients  of  V[7'']  and  [v  in  ll^csc 


42  THE    ADJUSTMENT    OF    OBSERVATIONS 

equations  for  values  of  n  from  2  to  100.     (See  Appendix,  Tables 

II,  III.)*  

If  Bessel's  formula  is  used,  compute  first  [2/-],  then  V['z/"]  can 
be  taken  from  a  table  of  squares  closely  enough.  This  square- 
root  number  multiplied  by  the  number  in  Table  II  correspond- 
ing to  the  given  value  of  n  gives  the  p.  e.  sought.  If  Peters' 
formula  is  used,  multiply  the  sum  of  the  residuals,  without  regard 
to  sign,  by  the  numbers  in  Table  III  corresponding  to  the 
argument  n. 

34.    Control  of  [^'^].  — A  control  is  afforded  by  the  deriva- 
tion of  [^''"']  from  the  observed  values  and  the  arithmetic  mean 
directly. 
We  have 

^1  =  ^0  —  ^v 
v^^  =  x-Q  —  Mj, 


^«  =  x^  -  M^. 
Square  and  add, 

\y'--\=nxi-2x,{M-\  +  [W\ 

=  [AP]-[M-]x,;  (i) 

since  w-Vo  =  {M\ 

The  values  of  HP  may  be  found  from  a  table  of  squares  or 
from  Crelle's  tables,  or,  if  the  numbers  J/ are  large,  an  arithmo- 
meter, or  machine  for  multiplying  and  dividing,  may  be  employed 
with  advantage. 

35.  Approximate  Method  of  Finding  the  Precision.  —  A 
connection  between  the  p.  e.  of  a  single  observation  and  the 
greatest  error  committed  in  the  series  may  be  established  ap- 
proximately by  the  aid  of  the  principle  proved  in  Arts.  22-23. 
There  we  saw  that  in  a  large  series  the  actual  errors  may  be  ex- 
pected to  range  between  zero  and  4  or  5  times  the  p.  e.  of  an 
observation.  If,  then,  we  find  from  the  observations  a  p.  e.  of 
an  amount,  say,  r,  we  may  assert  that  the  greatest  actual  error  is 
*  First  published  in  the  Analyst,  Des  Moines,  la.,  May,  1882. 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         43 

not  likely  to  be  more  than  5  r.  The  probability  of  its  being  as 
large  as  this  is  only  about  j  o\)  q- 

The  same  princii)le  will  enable  us  to  estimate  roughly  the  p.  e. 
in  a  series  of  observations.  A  glance  at  the  measured  results 
will  show  the  largest  and  smallest,  and  their  difference  may  be 
taken  as  the  range  in  the  results,  and  half  the  difference  as  the 
maximum  error.  Hence,  since  in  an  ordinary  series  of  from  25 
to  100  observations  the  maximum  error  may  be  expected  to  be 
from  3  to  4  times  the  p.  e.,  we  may  take  the  p.  e.  to  be  from  \  to 
i  of  the  range  of  tJie  errors  of  obscri'atioji. 

The  probable  error  so  estimated  is,  however,  rather  untrust- 
worthy, as  it  depends  upon  but  two  of  the  residuals  instead  of 
all  of  them.  Moreover,  these  are  the  two  residuals  which  cor- 
respond to  the  extreme  observations,  about  which  there  is  fre- 
quently a  reasonable  doubt  as  to  whether  they  should  not  be 
rejected. 

36.    Ex.  —  We  shall  now  apply  the  preceding  formulas  to  the  example  in 
Art.  27  to  find  the  p.  e.  of  the  arithmetic  mean  and  of  a  single  observation. 
(1)    The  p.  e.  of  ilie  arithmetic  mean. 
These  we  may  find  in  two  ways  : 
{a)    From  the  sum  of  the  squares  of  the  residuals  (Art.  29), 


Y  n  («  —  I] 


_ .  /  0.2756 


or  from  Table  II  at  once 


=  0.026, 
Tq  =  0.6745  ^  0.026 
=  0.017; 

ro  =  0.525  X  0.033 
=  0.017. 


{h)    From  the  sum  [7/  of  the  residuals  (Art.  32) : 

The  multiplier  in  Table  III  corresponding  to  the  number  21  is  0.009. 

.•.  ru  =  2.02  X  0.009 
=  0.018. 

(2)    The  p.  e.  of  a  single  observation. 
From  Tables  II.  and  III.  directly  : 


44  THE    ADJUSTMENT    OF    OBSERVATIONS 

r=  0.525  X  0.151  =-  0.079, 
r  =  2.02    X  o  041  =  0.082. 

C/iecA  (a).  Nine  residuals  out  of  twenty-one  are  less  than  the  computed 
p.  e.  i  0.079",  whereas,  according  to  theory  (Table  I),  one-half  the  errors, 
or  10^  out  of  21,  should  be  less  than  J^  0.079". 

This  is  the  practical  way  of  using  the  check.  We  might  have  arranged 
the  residuals  in  order  of  magnitude  when  the  residual  0.09  will  be  found  to 
occupy  the  middle  place. 

Check  [p).   See  Art.  35. 

Range  =  0.22  +  0.23  =  0.45. 

,.  r=^5_o.o8. 
o 

The  values  found  by  the  different  methods  agree  reasonably  well. 

37.    The  Law  of  Error  Tested  by  Experience.  —  We  shall 

now  test  our  example  and  see  how  closely  it  conforms  to  the 
law  of  error,  and  hence  be  in  a  better  position  to  judge  of  how 
far  the  law  of  error  itself  is  applicable  in  practice.  This  is  the 
a  posteriori  proof  intimated  in  Art.  6  as  necessary  for  the 
demonstration  of  the  law. 

(i)  The  number  of  +  residuals  is  12,  and  the  number  of  — 
residuals  is  9. 

(2)  The  sum  of  the  +  residuals  is  1.03,  and  the  sum  of  the 
—  residuals  is  0.99. 

(3)  The  sum  of  the  squares  of  the  +  residuals  is  14 17,  and 
of  the  —  residuals  is  1339. 

(4)  The  p.  e.  of  a  single  observation  is  0.08.  To  find  the 
number  of  observations  we  should  expect  whose  residual  errors 
are  not  greater  than  o.  10,  we  enter  Table  I  with  the  argument 

-^—  =  1.25  and  find  0.60.      This  multiplied  by  21  gives  13  as 

the  number  of  errors  to  be  expected  not  greater  than  o.  10.  By 
actual  count  we  find  the  number  observed  to  be  14. 

To  find  the  number  to  be  expected  between  o.  10  and  0.20 

we  enter  the  table  with  the  argument  ^^^  =  2.50  and  find  0.91. 

.08 

From  this  deduct  0.60  and  multiply  the  remainder  by  21.     This 

gives  6.     The  number  observed  is  5. 


OBSERVATIOXS    OF    ONE    UNKNOWN    QUANTITY  45 

The  number  to  be  expected  over  0.20  is,  by  theory,  2.     The 
number  observed  is  2. 

The  preceding  resuhs  are  collected  in  the  following  table : 


Limits  of  Error. 

Number  of  Ekkors. 

Theory. 

Observation 

JT.                 S. 

0.00    to    O.IO 
o.io  to  0.20 
over  0.20 

13 

6 
2 

14 

5 
2 

Table  I,  it  will  be  remembered,  is  founded  on  the  supposition 
that  the  number  of  observations  in  a  given  set  is  very  large. 
In  our  example  the  number  is  only  21.  Perfect  accordance 
between  the  number  of  errors  given  by  theory,  and  the  number 
given  by  observation  is,  therefore,  not  to  be  expected. 

38.  Caution  as  to  the  Application  of  the  Test  of  Preci- 
sion. —  In  the  preceding  article  we  have  given  several  cautions 
with  regard  to  the  strict  application  of  the  law  of  error  in  practice. 
We  shall  now  perform  a  similar  service  for  the  test  of  precision, 
the  probable  error.  The  probable  error  of  an  observation,  or  of 
the  mean  of  a  series  of  observations,  has  been  defined  as  a 
measure  of  accuracy  or  in  other  words  of  uncertainty.  It  must 
be  kept  clearly  in  mind  that  it  is  a  measure  only  of  such  uncer- 
tainties as  are  due  to  accidental  errors,  and  has  no  necessary 
relations  to  systematic  or  constant  errors.  If  all  the  errors 
occurring  in  the  observations  are  of  the  accidental  class,  tlie 
probable  error  is  a  true  measure  of  accuracy.  If  systematic  or 
constant  errors  also  occur,  these  give  rise  to  errors  in  the  result, 
in  addition  to  those  arising  from  accidental  errors,  and  the  p.  e., 
therefore,  expresses  but  a  part  of  the  uncertainty  of  the  rcsuh. 
The  neglect  of  this  principle  has  led  in  many  cases  to  erroneous 
conclusions,  and  to  faulty  methods  of  observing  and  computing. 
These  in  turn  have  led  to  wholesale  condemnation  of  the  method 
of  least-squares, 


46  THE    ADJUSTMENT    OF    OBSERVATIONS 

39.  The  fact  that  the  computed  probable  error  is  independent 
of  the  constant  error  in  the  observations  may  be  shown  from 
the  formula  from  which  it  is  computed. 

In  the  derivation  of  the  p.  e.  from  a  series  of  n  observed 
quantities  M^,  M^,  ...  we  had  the  observation  equations. 

Xq  —  Ml  =  v^ , 

Xq  —  M2  =  '^2» 


Xq  —  M„  =  f  „ 


Also  /-^  =  p  V2  ^^ 


Now,  if  we  suppose  each  of  the  observed  quantities  to  be 
changed  by  the  same  amount  c,  which  may  be  of  the  nature  of 
a  constant  error  or  correction,  so  that  they  become  J/j  +  c, 
jlf^-^c,.  .  .  the  most  probable  value,  instead  of  being  x^,  will 
be  x^  +  c.     Also  since 

V  =  (A'o  +  c)-  {M  +  c) 
=  xo  -  M, 

the  residuals  will  be  the  same  as  before. 

Hence  ;^  is  unchanged,  and  we  see,  therefore,  that  the  p.  e. 
makes  no  allowance  for  constant  errors  or  corrections  to  the 
observed  quantity.  These  are  supposed  to  be  eliminated  or 
corrected  for  before  the  most  probable  value  and  its  precision 
are  sought. 

In  leveling,  if  the  same  line  is  run  over  in  duplicate  in  the 
same  direction,  a  good  agreement  may  be  expected  at  the 
several  bench-marks  where  comparisons  are  made.  The  p.  e. 
of  observation  will  consequently  be  small.  If  the  line  is  levelled 
in  opposite  directions,  experience  shows  that  the  agreement 
would  not  be  so  good.  The  p.  e.  would  be  larger  than  before. 
We  might,  therefore,  hastily  conclude  that  the  first  work  would 
give  the  better  result.  But  when  we  reflect  that  the  main 
differences  arise  from  such  causes  as  the  rising  or  settling  of 
rods  and  instruments,  the  refraction  of  light,  .  .  .  which  causes 


OBSERVATIONS    OF    OXE    UNKNOWN    QUANTITY        47 

are  less  likely  to  be  mutually  destructive  and  more  likely  to  be 
cumulative  if  the  lines  are  run  in  the  same  direction,  it  is  to  be 
expected  that  the  final  result  obtained  from  measurements  in 
opposite  directions  will  be  nearer  the  truth.  The  conclusion 
arrived  at  by  trusting  to  the  p.  e.  alone  would  be  illusory,  for 
the  constant  and  systematic  errors  in  levelling  are  in  general, 
especially  on  long  lines,  much  larger  than  the  accidental  errors, 
and  the  p.  e.  is  simply  a  measure  of  the  effect  of  errors  of  the 
latter  class. 

40.  Another  common  misapprehension  is  the  following : 
From  Art.  13  the  relation  between  the  p.  e  of  a  single  obser- 
vation r  and  the  p.  e.  of  the  mean  of  «  observations  r^  is 

r 

V« 

This  form.ula  shows  that  by  repeating  the  measurement  a  suffi- 
cient number  of  times  we  can  make  the  p.  e.  of  the  final  result 
as  small  as  we  please.  Nothing  would,  therefore,  seem  to  be  in 
the  way  of  our  getting  an  exact  result,  and  that  we  could  do  as 
good  work  with  a  rude  or  imperfect  instrument  as  with  a  good 
one  by  sufficiently  increasing  the  number  of  observations. 

The  fallacy  lies  in  the  implied  supposition  that  all  the  errors 
affecting  the  observations,  are  of  the  accidental  class.  It  is 
true  that  the  effects  of  errors  of  this  class  will  be  reduced  by 
increasing  the  number  of  observations  in  the  manner  indicated 
above,  but  experience  indicates  that  in  all  observations,  constant 
and  systematic  errors  are  present  as  well  as  accidental  errors, 
beins:  sometimes  so  small  as  to  be  discernible  only  after  the 
accidental  errors  have  been  greatly  reduced  by  many  repetitions 
of  the  observations,  and  in  other  cases  so  large  as  to  be  evident 
after  the  first  few  observations  have  been  taken.  The  repeti- 
tion of  observations  has  no  effect  whatever  in  eliminating  the 
constant  errors,  and  none  in  eliminating  the  systematic  errors, 
unless  the  conditions  under  which  the  observations  are  taken 
be  by  accident  or  design  so  changed  from  time  to  time  as  tu 


48  THE    ADJUSTMENT    OF    OBSERVATIONS 

reverse  the  signs  of  the  systematic  errors.  In  general  the 
repetition  of  observations  reduces  the  error  of  the  result  very 
rapidly  at  first,  while  the  effects  of  the  accidental  errors  still 
predominate  over  those  of  other  classes,  and  while  each  change 
of  one  unit  in  n  produces  a  relatively  large  change  in  V//.  As 
the  observing  is  continued,  the  V//  changes  more  and  more 
slowly,  and  an  elimination  of  the  remaining  accidental  errors 
from  the  mean  is  correspondingly  slow.  Much  more  important 
than  this,  however,  is  the  fact  that  a  point  is,  sooner  or  later, 
reached  in  the  repetition  of  observations  at  which  the  unelimi- 
nated  accidental  error  is  smaller  than  the  constant  error  in  the 
observations  and  mean.  Beyond  this  point  the  effect  of  further 
observations  is  simply  to  reduce  the  smaller  and  comparatively 
unimportant  accidental  error,  and  leave  the  larger  serious  con- 
stant error  absolutely  unchanged.  The  effect  of  indefinitely 
continuing  the  observations  is  to  make  the  combined  accidental 
and  constant  error,  the  total  error  of  the  result,  approach  very 
slowly  to  the  constant  error  as  a  limit  when  the  number  of 
observations  is  infinite. 

41.  It  has  been  erroneously  assumed  by  many  persons  that 
the  limit  of  accuracy  for  a  given  instrument  is  the  smallest 
magnitude  that  can  be  seen  with  it,  that  "what  cannot  be  seen 
cannot  be  measured."  The  limit  of  accuracy  beyond  which 
one  cannot  go  by  increasing  the  number  of  observations,  is 
fixed  by  the  constant  and  systematic  errors  as  indicated  above 
and  has  no  necessary  relation  to  the  power  of  the  instrument 
used. 

Three  illustrations  may  be  given  of  measurements  of  which 
the  errors  of  the  mean  are  smaller  than  the  smallest  quantity 
which  can  be  seen  with  the  instruments  used.  With  the  best 
cheodolites  now  in  use  in  the  Coast  and  Geodetic  Survey,  the 
mean  of  16  measures  of  a  direction  has  a  probable  error  in  gen- 
eral from  i  to  5  of  a  second  of  arc.  The  checks  which  are 
available  show  these  probable  errors  to  be  true  measures  of  the 
accuracy.     These  observations  are  made  with  a  telescope  with 


OBSERVATIONS    OF    ONE    UNKNOWN    OUANTITY 


49 


which  it  is  impossible  under  the  actual  conditions  of  observation 
to  see  a  rod  or  stripe  J^  to  ^  of  an  inch  wide  placed  a  mile  from 
the  instrument,  and  therefore,  subtending  an  angle  of  from  i  to 
i  of  a  second. 

With  the  zenith  telescope,  now  being  used  for  the  work  of 
determining  the  variation  of  latitude  under  the  direction  of  the 
International  Geodetic  Association,  the  probable  error  of  a 
single  observation  has  frequently  been  found  to  be  as  small  as 
±  o'.  lO,  which  is  a  small  fraction  of  the  width  of  the  line  used 
in  bisecting  the  stars  and  beyond  the  power  of  vision  of  the 
observer  as  aided  by  the  telescope.  The  probable  error  of  the 
mean  result  from  a  night's  work  is  very  much  smaller  than  this, 
±  o^04  or  less.  The  checks  obtained  from  combining  the 
work  of  different  observatories  show  that  these  probable  errors 
are  true  measures  of  the  accuracy,  or  in  other  words,  that  the 
constant  and  systematic  errors  are  extremely  small. 

In  the  precise  level  net  of  the  United  States,  there  are 
thousands  of  miles  of  leveling  with  the  new  Coast  and  Geodetic 
Survey  precise  level,  for  which  the  largest  correction  expressed 
in  millimeters  per  kilometer  arising  from  the  necessity  of  closing 
all  circuits,  the  most  severe  test  of  accuracy  which  can  be  apj^lied, 
is  Jj  of  a  millimeter  per  kilometer.  The  readings  in  this  kind  of 
leveling  are  made  on  a  direct  reading  rod  by  estimating  the 
position  of  each  of  three  cross-lines  in  the  telescope  as  seen 
projected  against  a  centimeter  graduation  on  the  rod.  Each 
reading  is  taken  to  the  nearest  millimeter  only.  It  is  abso- 
lutely impossible  to  see  so  small  a  magnitude  as  ^^^  millimeter 
on  the  rod  through  the  telescope  at  the  distance  at  which  the 
rod  is  ordinarily  placed,  yet  the  accidental  errors  are  reduced 
below  this  limit  for  a  whole  kilometer  involving  in  general  1 2  to 
14  sights. 

42.  The  proposition  that  one  cannot  measure  smaller  mag- 
nitudes than  can  be  seen  is  a  dangerous  error,  for  the  reason 
that  it  is  liable  to  lead  to  pr)or  habits  and  poor  methods  of  ob- 
servation.     The  student,  for  c.\;im])lc,  while  being  taught  to  use 


50  THE    ADJUSTMENT    OF    OBSERVATIONS 

the  zenith  telescope  for  latitude  observations,  on  being  warned 
that  he  must  be  extremely  careful  not  to  apply  any  longitudinal 
pressure  on  the  head  of  the  micrometer  screw,  will  sometimes 
experiment  for  himself  by  purposely  applying  such  a  light 
pressure  while  watching  the  bisection.  He  may  not  be  able  to 
see  any  change,  becomes  skeptical,  and  thereafter  is  slovenly  in 
his  handling  of  the  instrument.  Similarly,  the  observer  with  a 
precise  level,  on  being  told  that  an  extremely  small  amount  of 
unequal  heating  of  the  level  vial  will  cause  a  bubble  to  travel,  and 
introduce  an  error  into  the  results,  will  experiment  and  convince 
himself  that  the  movement  of  this  character  ordinarily  encoun- 
tered under  actual  field  conditions  is  on  an  average  smaller 
than  he  can  see.  He  may  then  use  a  method  for  years  which 
is  radically  defective  in  not  guarding  against  this  source  of 
error.  Although  the  motion  of  the  bubble  may  be  too  small  to 
be  visible,  yet  the  principal  error  in  his  work  may  be  due  to 
this  cause. 

43.  As  an  example  of  a  case  in  which  the  constant  errors  are 
so  large  that  little  is  gained  in  accuracy  after  the  first  few  obser- 
vations, the  determination  of  absolute  declinations  and  right  ascen- 
sions may  be  cited.  With  the  meridian  circle  Professor  Rogers 
found  the  p.  e.  of  a  single  complete  observation  in  declination 
to  be  ±  o".T)6,  and  the  p.  e.  of  a  single  complete  observation  in 
right  ascension  for  an  equatorial  star  to  be  ±  o'.026.  He  says  : 
"  If,  therefore,  the  p.  e.  can  be  taken  as  a  measure  of  the 
accuracy  of  the  observations,  there  ought  to  be  no  difficulty  in 
obtaining  from  a  moderate  number  of  observations  the  right 
ascension  within  0''.02  and  the  declinations  within  o".2.  Yet  it 
is  doubtful,  after  continuous  observations  in  all  parts  of  the  world 
for  more  than  a  century,  if  there  is  a  single  star  in  the  heavens 
whose  absolute  co-ordinates  are  known  within  these  limits."  * 
The  explanation  is,  as  intimated,  that  constant  errors  are  not 
eliminated  by  increasing  the  number  of  observations.  Acci- 
dental errors  are  eliminated  by  so  doing. 

*  Froc.  Ainer.  Acad.  Sci.,  1878,  p.  174. 


OBSERVATIOXS    OP    OXE    UXKXOWX    OUAXTlTV 


5' 


44.  There  is  a  common  idea  that  if  we  have  a  poor  set  of 
observations,  good  results  can  be  derived  from  them  by  adjusting 
them  according  to  the  method  of  least  squares,  or  that,  if  work 
has  been  coarsely  done,  such  an  adjustment  will  bring  out  results 
of  a  higher  grade.  The  method  of  least  squares  is  a  method  of 
computation,  not  of  observation,  which  serves  merely  to  aid  the 
computer  to  secure  in  his  computed  results  the  highest  grade  of 
accuracy  possible  from  a  given  series  of  observations,  but  it  can- 
not increase  the  accuracy  of  observations  already  taken.  The 
observations  fix  an  absolute  limit  to  attainable  accuracy.  The 
computer  may  approach  this  limit  more  or  less  closely 
according  to  his  skill,  but  cannot  pass  it.  A  third  puzzle  in  con- 
nection with  probable  error  may  be  mentioned.  It  may  happen 
that  the  value  obtained  of  the  p.  e.  is  numerically  greater 
than  that  of  the  observed  quantity  itself.  It  is  then  a 
question  whether  in  subsequent  investigations  we  should  use 
the  value  of  the  observ^ed  quantity  as  found,  or  neglect  it. 
This  depends  on  circumstances.  It  is  ever  a  principle  in 
least  squares  to  make  use  of  all  the  knowledge  on  hand  of 
the  point  at  issue.  If  we  have  strong  a  priori  reasons  for 
expecting  the  value  zero,  it  would  be  better  to  take  this  value. 
Thus,  if  we  ran  a  line  of  levels  between  two  points  on  the  sur- 
face of  a  lake,  we  should  expect  the  difference  of  height  to  be 
zero.  If  the  p.  e.  of  the  result  found  were  greater  than  the 
result  itself,  it  would  be  allowable  in  this  case  to  reject  the  deter- 
mination. On  the  other  hand,  when  we  have  no  a  pnori  knowl- 
edge, as  in  determinations  of  stellar  parallax,  for  example,  if  the 
p.  e.  of  the  value  found  were  in  excess  of  the  value  itself,  as  is 
sometimes  the  case,*  we  could  do  nothing  but  take  the  value 
resulting  from  the  observations,  unless,  indeed,  it  came  out  with 
a  negative  sign,  and  then  its  untrustworthy  character  would  be 
evident. 

45.  Systematic  Error.  —  References  to  systematic  error  in 
the  preceding  articles  lead  us  to  notice  an  example  or  two  (»f 

*  See,  for  example,  Newcomb,  /Isiroiioiiiy,  app.  vii. 


52  THE    ADJUSTMENT    OF    OBSERVATIONS 

the  detection  and  treatment  of  this   great   bugbear  of    obser- 
vation. 

We  suspect  the  presence  of  systematic  errors  in  a  series  of 
observations  from  finding  that  the  residuals  do  not  bear  the  rela- 
tions to  each  other  that  they  would  if  the  errors  were  all  of  the 
accidental  class,  or  from  other  departures  of  the  results  from  the 
laws  which  they  would  follow  if  the  errors  were  all  accidental. 

Sometimes  the  sources  of  error  are  detected  without  much 
trouble.  Thus,  in  measuring  an  angle  with  a  theodolite,  if  the 
instrument  is  placed  on  a  stone  pillar  firmly  embedded  in  the 
ground,  the  range  in  results,  if  targets  are  the  signals  pointed  at, 
would  not  usually  be  over  lo"  in  primary  work  ;  and  on  reading 
to  a  number  of  signals  in  order  round  the  horizon,  the  final  read- 
ing on  closing  the  horizon  would  be  nearly  the  same  as  the 
initial  reading  on  the  same  signal.  If,  next,  the  instrument  were 
placed  on  a  wooden  post,  and  readings  made  to  signals  in  order 
round  the  horizon  in  the  same  way  as  before,  the  final  reading 
might  differ  from  the  initial  by  a  large  amount.  The  observa- 
tions might  also  show  that  the  longer  the  time  taken  in  going 
around,  the  greater  the  resulting  discrepancy.  The  natural 
inference  would  be  that  in  some  way  the  wooden  post  had  to  do 
with  the  discrepancy  in  the  results.  In  an  actual  case  *  of  this 
kind,  examination  showed  the  change  to  be  most  uniform  on  a 
day  when  the  sun  shone  brightly.  Measurements  were  then 
made  at  night,  using  lamps  as  signals  on  the  distant  stations, 
and  the  same  change  was  observed,  only  it  was  in  the  opposite 
direction. 

The  effect  on  the  value  of  an  angle  of  this  twist  of  station, 

*  At  U.  S.  Lake  Survey  station  Bruld,  Lake  Superior,  many  observations 
were  taken  during  both  day  and  night  in  July,  1S71,  to  determine  the  rate  of 
twist  of  center-post  on  which  the  theodoHte  used  in  measuring  angles  was 
placed.  The  conclusion  arrived  at  was  that  "during  a  day  of  uniform  sun- 
shine and  clear  atmosphere  this  twist  seemed  to  be  quite  regular,  and  at  the 
rate  of  about  one  second  of  arc  per  minute  of  time,  reaching  a  maximum 
about  7  p.  M.  and  a  minimum  about  7  A.  M.,  during  the  month  of  July.  On 
partially  cloudy  days  there  was  no  regularity  in  the  twist,  being  sometimes 
in  one  direction  and  again  in  the  opposite." 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY        53 

assuming  it  to  act  uniformly  in  the  same  direction  during  the 
time  of  observation,  can  be  eUminated  by  the  method  of  obser- 
vation :  first,  reading  to  the  signals  in  one  direction,  and  then 
immediately  in  the  opposite  direction,  and  calling  the  mean  of 
the  difference  of  the  two  sets  of  readings  a  single  value  of  the 
angle.  So  also  in  azimuth  work  the  mean  of  the  difference  of 
the  readings,  star  to  mark,  and  mark  to  star,  gives  a  single  value 
free  from  station-twist. 

This  mode  of  procedure  is  in  accordance  with  the  general 
principle  to  eliminate  a  systematic  error,  when  possible,  by  the 
method  of  observation,  rather  than  to  compute  and  apply  it. 

46.  The  effort  to  avoid  systematic  error  causes  in  general  a 
considerable  increase  of  labor,  and  sometimes  this  is  very  marked. 
For  example,  in  the  micrometric  comparison  of  two  line  meas- 
ures belonging  to  the  U.  S.  Engineers,  the  results  found  by 
different  observers  showed  large  discrepancies.  The  micro- 
meter microscopes  used  were  of  low  power,  with  a  range  of 
about  one  mm.  between  the  upper  and  lower  limits  of  distinct 
vision.  Examination  showed  that  the  discrepancies  arose  mainly 
from  focusing,  each  observer's  results  being  tolerably  constant 
for  his  own  focus.  As  the  value  of  a  revolution  of  the  micro- 
meter screw  entered  into  the  reduction  of  the  comparison  work, 
and  as  this  value  was  obtained  from  readings  on  a  space  of 
known  value,  error  of  focusing  entered  from  this  source.  Hence 
a  value  of  the  screw  had  to  be  determined  from  a  special  set  of 
readings  taken  at  each  adjustment,  and  this  value  used  in  redu- 
cing the  regular  observations  made  with  the  same  focus.  Had 
the  microscopes  been  of  high  power,  it  would  have  been  sufficient 
to  determine  the  value  of  the  screw  once  for  all,  since  the 
error  arising  from  change  of  focus  could  have  been  classed  as 
accidental. 

In  trying  to  avoid  or  eliminate  systematic  error,  the  observer 
will,  as  he  gains  in  experience,  take  precautions  which  would  at 
first  seem  to  be  almost  childish.  Good  work  can  only  be  had  at 
the  co.st  of  eternal  viirilance. 


54  THE    ADJUSTMENT    OP    OBSERVATIONS 

B.    Observed   Values  of  Different  Quality. 

47.   The  Most  Probable  Value ;  the  Weighted  Mean.  —  It 

has  been  shown  in  Art.  12  that  if  the  directly  observed  values 

J/,,  i/^,  .  .  .  M^  of  a  quantity  are  of  different  quality,  the  most 

probable  value  is  found  by  multiplying  each  residual  error  of 

observation  by  the  reciprocal  of  its  p.  e.,  and  making  the  sum  of 

the  squares  of  the  products  a  minimum  ;  that  is,  with  the  usual 

notation, 

or  -^  +  ^+---+^  =  a  mm.,  (i) 

By  differentiation  and  reduction. 

We  have,  therefore,  the  equivalent  rule  : 

If  the  observed  values  of  a  quantity  are  of  diffejrnt  quality, 
the  most  probable  value  is  found  by  multiplying  each  observed 
value  by  the  reciprocal  of  the  square  of  its  p.  e.,  and  dividing  the 
sum  of  the  products  by  the  sum  of  the  reciprocals. 

The  form  of  the  expression  for  ,r„  suggests  another  standpoint 
from  which  to  consider  it.     Let  /,,  p.^,  .   -   -  pn  be  the  numerical 

parts  of— ^  »  —  »  •  •  •    —^y   such  that  each  is  of  the  type 


(unit  of  measure)^ 

P  = ;5 5 

then  equation  (3)  may  be  written, 

^o-j^  (4) 

Also,  since  _ ^  >   _  >  •  •  •    — ^,  are  similarly  involved  in  the  nu- 

r'     r  r " 

1        2  " 

merator  and  denominator  of  the  value  of  x^,  this  value  will  re- 
main the  same  if  /j,  p,^,  .  .  .  /„  are  taken  any  numbers  whatever 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY  55 

in  the  same  proportion  to^7»  — 5>  •  •  •  -^J  that  is,  if  A,  A,, 
.  .  .  /„  satisfy  the  relations 

A  =7-2'        /'2=-l'      •     •     •'     ^'=-7  (S) 

'1  '2  'n 

where  r  is  an  arbitrary  value  of  the  probable  error  corresponding 
to  the  arbitrarily  assumed  unit  weight.  The  numbers  p^,  p.^,  p^, 
■  .  .  p„  are  called  the  weigJits,  or,  better,  the  combining  weights 
of  the  observed  values,  and  the  mean  value  .r^  is  called  the 
weighted  mean. 

The  expression  [/-^^J/C/]  can  now  be  put  into  words  as  fol- 
lows : 

If  the  observed  values  of  a  quantity  are  of  different  zveights, 
the  most  probable  value  is  foujid  by  multiplying  each  observed 
value  by  its  zveight,  and  dividing  the  sum  of  the  products  by  the 
sum  of  the  weights. 

In  Arts.  13,  14,  as  indicated  in  the  expressions  for  the  p.  e.  of  a 
single  observation  and  of  the  mean,  it  was  shown  that  for  obser- 
vations of  equal  precision,  or,  in  other  words,  of  equal  weight, 
the  p.  e.  of  a  mean  of  n  observations  is  to  the  p.  e.  of  a  single 
observation  as  i  is  to  V;/.  By  comparison  of  these  with  the  ex- 
pressions 

Pi   =—2^         /'2  =  -^  '••     •'/>«  =  — 2  ' 
'  1  '■>  '  n 

it  may  be  seen  that  the  meaning  of  weight  p^  assigned  to  an  ob- 
servation is  that  it  has  the  same  degree  of  accuracy  as  the  mean 
of /j  observations  of  unit  weight.  In  combining  observations  it 
is  treated  accordingly.  With  this  understanding,  it  is  evident 
that  to  combine  observed  values  J/,,  M,^,  M ^,  .  .  .  J/„,  of  which 
the  weights  are  /,,  />.„  />,,,  ,  .  .  /„,  one  should  proceed  as  if  J/, 
were  the  mean  of  p^  separate  observations  of  unit  weight,  J/, 
of  p.,  observations  of  unit  weight,  and  so  on.  The  arithmetic 
mean  of  the  hypothetical  [/^]  observations  would  evidently  be 
[/'il/]/[/'],  and  this  is  precisely  the  form  used. 


56  THE    ADJUSTMENT    OF    OBSERVATIONS 

If  the  observed  values  M  are  numerically  large  we  may 
lighten  the  numerical  work  by  finding  x  the  method  of  Art.  26. 
Proceeding  as  there  indicated,  we  have 

=  X'  +  x"  suppose. 

48.  Reduction  of  Observed  Values  to  a  Common  Stan- 
dard. —  The  principle  of  the  weighted  mean  is  evidently  an  ex- 
tension of  that  of  the  arithmetic  mean,  as  was  pointed  out  long 
ago  by  Cotes,  Simpson,  and  others.  It  merely  amounts  to  finding 
a  mean  of  several  series  of  means,  the  unit  of  the  measure  being 
the  same  in  each.  As  soon,  therefore,  as  results  of  different 
weights  are  changed  into  others  having  a  common  standard  of 
weight,  the  rules  for  combining  and  finding  the  precision  of  ob- 
served quantities  of  the  same  weight  can  be  applied  to  weighted 
quantities. 

This  change  we  are  enabled  to  make  by  means  of  the  relation 
(5),  Art.  47,  which  may  be  written, 

r  r  r 

r^  =  -1=  .     r._  =  -^  '    •  •  •  '    r„  =  -=  • 

ypi         yp2  '^pn 

Now,  since  r,,  ;;,  .  .  .  r„  are  the  p.  e.  of  J/j,  3f.,,  .  .  .  M„, 
the  p.  e.  of  M^  V/,,  M,  V/^,  .  .  .  M„  \f/>~  would  each  be  the 
same  quantity  r. 

Hence,  if  a  series  of  observed  values  M^,  M ,^^  .  .  .  A/„ /lave  t/ie 
weigJits  p^,  p.„  .  .  .  />„,  t/uy  are  reduced  to  the  same  standard  bv 
miiltiplying  by  \^,   'V^,   .   .    .    V/,^  respectively. 

For  example,  given  the  observ^ation  equations, 

X  —  My  —  Vy  weight  p^ , 
X  —  M^  =  v^^  weight  p^_ , 


X  —  M„  =  Vn  weight  ^„, 
to  find  the  most  probable  value  of  x. 


OBSERVATIONS   OP    ONE    UNKNOWN    QUANTITY        57 

Reducing   to   the    same   standard   of  weight,   we    have    the 
equations, 


and  the  most  probable  value  of  x  is  found  by  making 

(VK^i)'  +  (^72-^2)'  +  •  •  •  +  {^fnV„y  =  a  min. ; 
that  is,  by  making 

Reducing  this  equation,  we  find,  as  before, 


Xq 


[P] 


49.   Control  of  the  Weighted  Mean.  —  Eq.  4,  Art.  47,  may 

be  written, 

[piq  =  o. 

The  error  of  any  assigned  value  of  x^  is  evidently  [//]  /  [/>], 

in  which  the   I's   are  residuals   corresponding  to  the  assumed 

value  of  x^^.     If  this  error  is  less  than  one-half  unit  in  a  given 

decimal  place,  the  assumed  value  is  correct  to  that  place. 

Ex.  —  Find  the  most  probable  value  of  the  velocity  of  light  from  the  fol- 
lowing determinations  by  Fizeau  and  others : 

298,000  kil.  -j-  1000  kil. 
298.500   "     ^  1000  " 
299,990  "     ^    200  " 
300,100   "     :^  1000   " 
299,930  "     ^     100  " 

[Amer.Jour.  Set.,  vol.  xix.) 

The  weights,  being  inversely  as  the  squares  of  the  probal)le  errors,  are  as 
the  numbers  i,  i,  25,  i,  100.    (Art.  47.) 

To  avoid  handling  such  large  numbers  as  the  M's,  wc  may 
assume  a  value  X'  for  ,1-,  by  inspection,  say  299,900,  and  tlK-ii 
proceed  as  follows : 


58 


THE    ADJUSTMENT    OF    OBSERVATIONS 


/ 

P 

pi 

+  1900 
+  1400 

-  90 

—  200 

-  30 

I 
I 

25 

I 
100 

+  1900 
+  1400 

—  2250 

—  200 
-3000 

128 

—  2150 

The  correction  yf'  to  the  assumed  vahie 


-  (-  2150) 
128 


+  17, 


and  the  weighted  mean  =  299,017. 

[It  is  much  more  important  in  computations  to  keep  the 
numbers  as  small  as  possible  than  to  avoid  minus  signs.  Such 
a  procedure  saves  time.  It  is  almost  impossible  to  arrange 
computations  so  that  the  computer  will  not  be  obliged  to  watch 
the  signs.  He  can  watch  many  minus  signs  as  well  as  a  few.  In 
the  long  run,  rapidity  depends  rather  upon  the  number  of  signi- 
ficant figures  used  in  the  quantities  handled.] 

50.  The  Precision  of  the  Weighted  Mean.  —  Since  the 
weighted  mean  x\^  is  the  arithmetic  mean  of  [/>]  observations  of 
the  unit  of  weight,  its  weight  is  [/].  Hence  the  p.  e.  r^  of  x^ 
is  found  from 

yS. 

2 

where  r  is  the  p.  e.  of  an  observation  of  the  unit  of  weight 
(standard  observation). 

According  to  Art.  48,  the  value  of  r  may  be  found  by  writing 
V^T'  ,  V^j'Z'o,  ...  for  z\y  V.,,  .  .  .  m.  the  formulas  derived 
for  observations  of  the  same  weight.  Hence,  substituting  in 
Bessel's  and  in  Peters'  formulas.  Arts.  29  and  32,  we  have 


•6745  v/-^^^ 

^  n  —  \ 


or  r=  0.8453 


'^v 


and  therefore 


^n(f 


OBSERVATIONS    OF    OXE    UNKNOWN    QUANTITY 


59 


''o  =  -6745  \ 


rw^ 


or  ro=  0.8453 


sTpv 


[p\{n~i)  "  '^^  ^[p]n{n-i) 

These    expressions    reduce    to   those  for  the  arithmetic  mean 
where  the  observed  values  are  of  the  same  weight  by  putting 

[/]  =  ^ip- 

Ex.  —  The  linear  values  found  for  the  space  0.00"  to  0.05"  of  inch  [ab] 
on  the  standard  steel  foot  i  F.  of  the  G.  T.  Survey  of  India  were  as  fol- 
lows: 0.050027",  0.049971",  0.050019",  0.050079",  0.050021",  0.05001 1".  The 
numbers  of  measures  in  these  determinations  were  6,  6,  15,  15,  8,  8,  respec- 
tively. 

Taking  the  numbers  of  measures  as  the  weights  of  the  respective  deter- 
minations, required  the  most  probable  value  of  the  space  and  its  p.  e. 

The  direct  solution  presents  no  difficulty.  The  value  of  x^  may  be  found 
as  in  Ex.  Art.  49,  and  thence  the  residuals  v.  The  p.  e.  follows  from  the 
formulas  of  Art.  50. 

Assimie  X'  =  0.049971. 


p 

I 

Pl 

V 

v^ 

pv^ 

6 
6 

15 
15 

8 

+  .000056 
+     0 
+    48 
+    loS 

+     50 
-f     40 

+  .000336 
+      0 

+    720 
+   1620 
+    400 
+    320 

+  .000003 

+    59 
^           II 
49 
+     9 
+     19 

.000000000009 

3481 

121 

2401 

81 

361 

.000000000054 
20S86 

1815 

36015 

648 

2888 

58 

+  -003396 

000000062306 

...  .v"  =  '-.^  =  +0.000059,   /-  =  0.6745  y  7~^^x  =  0.000075", 

and  x,=X'  +  x"=  0.050030,  ro=  o-6745  \/ (^7177)1;,]=  °-°°°°'°' 


Hence, 


Xq  =  0.050030"  ±  o.ooooio. 


51.  In  the  above  example  an  important  practical  point 
occurs,  and  one  often  overlooked.  The  p.  e.  is  not  computed 
from  the  original  observations,  but  from  these  observations 
grouped  in  six  sets  of  means.     These  means  we  have  treated 


6o  THE    ADJUSTMENT  OF    OBSERVATIONS 

as  if  they  were  original  observations  of  certain  weights.  Had 
the  original  observations  been  accessible  we  should  have  used 
them,  and  would  most  probably  have  found  a  different  value 
of  the  p.  e.  from  that  which  we  have  obtained,  some  of  the 
facts  having  been  partially  concealed  by  the  process  of  taking 
the  means  of  the  separate  groups. 

In  good  work  the  difference  to  be  expected  between  the 
value  of  the  p.  e.  found  from  the  means  and  that  found  from 
the  original  observations  would  be  small.  Still,  whenever 
there  is  a  choice,  the  p.  e.  should  always  be  deduced  from  the 
original  observations  rather  than  from  any  combinations  of 
them. 

The  weighted  mean  value  -r^  would  evidently  be  the  same 
whether  computed  from  the  partial  means  or  from  the  original 
observations. 

Obseived   Values  Multiples  of  the   Unknown. 

52.  Let  the  observed  values  M^,  M^,  .  .  .  J-/,,  be  multiples 
of  the  same  unknown  X ;  that  is,  be  of  the  form  a^X,  a.,X,  .  .  . 
a,^X,  where  a^,  a.,,   .  .  .  a„  are  constants  given  by  theory  for 

each  observation.     The  values  — \  -^,  .   .  .  — ~  of  X  may  be 

regarded  as  directly  observed  values  of  unequal  weight.     If  r  is 

the  p.  e.  of  an  observation,  that  is,  of  M ^,  AT ,^,   .   .   .,   then, 

1\I        r         M       r 
since  the  p.   e.   of    — ^  is  — ,  of  — ?  is  — ,  .   .   .   the  weights  of 
a^       a^  «2        a.^ 

these  assumed  observations  are  proportional  to  a^,  a^.    .    .    . 

Hence,  taking  the  weighted  mean, 


tZi"  4-  a^^-h  •  •  •  +  a„2 
_[aM] 


Also,  since  [<^']  is  the  weight  of  X, 


OBSERVATIOXS    OF    ONE    UNKNOWN    QUANTITY         6i 


Ex.  —  To  test  the  power  of  the  telescope  of  the  great  theodolite  (3  ft.)  of 
the  English  Ordnance  Survey,  and  find  the  p.  e.  of  an  observation,  a 
wooden  framework  was  set  up  12,462  ft.  distant  from  the  theodolite  when  at 
station  Ben  More,  Scotland.  It  was  so  arranged  that  when  projected 
against  the  sky  a  fine  vertical  line  of  light,  the  breadth  of  which  was  regu- 
lated by  the  sliding  of  a  board,  was  shown  to  the  observer.  The  breadth 
of  this  opening  was  varied  by  half-inches  from  ih  in.  to  6  in.  during  the 
observations,  which  were  as  follows:* 


No.  OF  Obser- 
vations. 

Width. 

Side  of  Opening. 

Mean  of  Microscofk 
Readings. 

I 
2 

3 

4 
5 
6 

7 
8 

9 

ID 

6.0 

5-5 
5.0 

4-5 
4.0 

3-5 
3-0 

2-5 

2.0 
1-5 

(left 
\  right 
(left 
I  right 
(left 
1  right 
(left 
1  right 
(left 
1  right 
(left 
\  right 
(left 
\  right 
(left 
1  right 
(left 
(  right 
jleft 
)  right 

(  28.00 
1  37.50 
1  28.50 
(  3700 
(  29.16 

(37-16 
(  30.16 
I  36.66 

30.50 
(37-16 
(31.16 
1  37-00 
S  32-66 

\  36-83 
J  33-50 
1 36.83 

)  3383 
1 37-00 

(37-16 

Let  X  =  the  most  probable  value  of  the  angle  subtending  an  opening  of 
I  inch.     1  hen  we  have  the  observation  equations, 

6X  -  9.50  =  -y,  3.5  X  -  5.84  =  7/0 

5.5  A' -  8.50  =  t/j  3^- 4-17  =  'Z'7 

SX  -  8.00  =  V3  2.5  X  -  3.33  =  T/g 

4.5  X  -  6.50  =  7'4  2X  -  3.17  =  t/9 

4X  -  6.66  =  7/,,  1.5  A'  -  1.66  =  7/,„ 

From  the  preceding  we  have  for  the  individual  values  of  X  and  their 

weights, 

*  Account  of  the  Principal  Triangulation,  pp.  54,  55. 


62  THE    ADJUSTMENT    OF    OBSERVATIONS 

X  =  1.58,     weight  6^ 
X  =  1.55,     weight  5.52. 


.  ,  ^  ,               9.50  X  52  +  8.50  X  5.5'+- .  • 
.-.  weighted  mean  =  ^-^ 52  ^  ^.^2  ^  .  .  . 

=  i-55> 
or  making  the  sum  of  the  squares  of  the  residuals  %>  a  minimum,  that  is, 
(6  X  -  9.50)2  +  ^^^  X  -  8.50)2  +  .  .  .  =  a  min., 

we  find  by  differentiation  that 

-V=i.55, 
as  before. 

The  practical  rule  following  from  either  method  is  the  same,  and  may  be 

stated  thus:  Multiply  each  observation  equation  by  the  coefficient  of  X  in 

that  equation,  and  add  the  products.     The  resulting  equation  gives  the  value 

of  ^. 

Precision  of  a  Linear  Function  of  Independently  Observed 

Quantities. 

53.  Suppose  that  there  are  ;/  independently  observed  quanti- 
ties yl/,,  M^y,  .  .  .  whose  m.  s.  e.  are  fi^,  /a^,  .  .  .  respectively, 
to  find  the  p.  e.  r  of  F  where 

F  =  a^M^  +  a^M^  +  •  •  •  +  ajl„,  (i) 

a^,  a^,  .  .  .  a^  being  constants. 

If  Aj,  A,„  .  .  .  denote  the  errors  of  M^,  M^,  .  .  .  we  shall 
have  the  true  value  7"  of  i^  by  writing  M^  -f  A^,  M.^  -f  A^,  .  .  . 
for  J/j,  M^,  ...  in  the  above  expression  for  F ;  that  is, 

T  =  a,  (Jf  1  +  Aj)  +  a,  (M,  +  A,)  +  •  .  .  +  a„  (.1/.  +  A„). 

Call  A  the  error  of  F;  then,  since  7"  =  7^  +  A,  we  have 

A   =   ai\    +   <?2^2    +     •    •    •     +   '^'n^n^ 

and  .-. 

A-  =  rtf  Aj^  +  (7^,^A/  +  .  •  .  +  2  a^a.^^^^.y  +  •  •  • 

Let  the  number  of  sets  of  Jf,,  Af.,,  .  .  .  required  to  find  T  be 
n,  and  suppose  A^  summed  for  all  the  sets  of  values  of  A^,  A^, 
.  .  .  and  the  mean  taken,  then  attending  to  Art.  13, 

fjij?  =  a^Vi^  +  <^2"A*2^  +  •  •  ■+  2  a^a^      ^        -h  .  .  .  (2) 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY        63 

In  forming  all  possible  values  of  A,A.„  A.A^,  .  .  .,  the  num- 
ber of  \alues  being  very  large,  there  will  probably  be  about  as 
many  +  as  —  products  of  each  form,*  and  we  therefore  assume 

[A^AJ  =  [A3A3]  =  .  .  .  =  o. 
Hence 

M^  =  [(7VJ,  (3) 

and 

rr  =  {irr]. 

Ex.  I.  —  The  Keweenaw  Base  was  measured  with  two  measuring  tubes 
placed  end  to  end  in  succession.  Tube  i  was  placed  in  position  967  times, 
and  tube  2.  966  times.  Given  the  p.  e.  of  the  length  of  tube  i  =  ^  0.00034", 
and  of  tube  2  =  J^  0.00037",  find  the  p.  e.  in  the  length  of  the  line  arising 
from  the  uncertainties  in  the  length  of  the  tubes. 

[p.  e.  from  tube  i  =  967  x  0.00034  =  0.329" 
p.  e.  from  tube  2  =  966  x  0.00037  =  0.357" 
.".  p.  e.  of  line      =  \/o.329^  +  0.357^ 
=  0,485".] 

Ex.  2.  —  In  the  Keweenaw  Base  the  p.  e.  of  one  measurement  of  94  tubes, 
deduced  from  the  discrepancies  of  six  measurements  of  these  94  tubes,  was 
found  to  be  0.03".  Show  that  the  p.  e.  in  the  length  of  the  line  of  1933  tubes 
arising  from  the  same  causes  may  be  estimated  at  ^  0.136". 

[p.  e.  of  I  measurement  of  i  tube  =  ■  '_ 

V94 

p.  e.  of  base  of  1933  tubes  =     '_  V1933 

V94 
=  i  0.136.] 

Attention  is  called  to  these  two  problems,  from  the  impor- 
tance of  the  principles  illustrated.  In  Ex.  i  the  p.  e.  of  a  tube 
was  multiplied  by  the  whole  number  of  tubes  to  find  the  p.  e.  of 
the  base  from  that  cause,  for  the  reason  that  with  whatever 
error  the  tube  is  affected,  it  is  cumulative  throughout  the 
measurement. 

In  Ex.  2  the  p.  e.  of  one  tube  is  multiplied  by  the  square  root 
of  the  number  of  tubes,  because  each  mcasiu'cmcnt  is  inde]K'n- 
dent  of  every  other,  and  the  errors  are  as  likely  to  be  in  excess 
as  in  defect,  and,  therefore,  may  be  expected  to  destroy  one  an- 
other in  the  final  result. 

*  See  footnote  to  Art.  20. 


64  THE    ADJUSTMENT    OF    OBSERVATIONS 

54.  If  the  function  F  whose  p.  e.  is  required  is  not  in  the 
linear  form,  we  first  reduce  it  to  that  form.     Thus,  if 

the  true  value  T  oi  F  will  result  if  we  write  M^  +  dM^,  M^  + 
dM^^  ...  for  J/j,  M,,  .  .  .  the  differentials  representing  the 
errors  of  these  quantities.     Then 

T=f(M  +  dAI,  if.  +  dM.^,  .  .  .)• 
Expanding  by  Taylor's  theorem,  and  retaining   only  the   first 
powers  of  the  small  quantities  dJll^,  dJl/,,  .   .  .,  we  have, 

or 

Error  ot  F  =  a.dM^  +  aJM^  -{-...+  aJAI^  (i) 

where  «i  =  sIT;  "^^  =  SmJ'  "  '  '  ^"  "  8M:* 

This  expression  is  of  the  same  form  as  (i).  Art.  53,     Hence, 
rjr  =  flf/'i^  +  a.-r,^  +  •  •  •  +  ajrj  =  [a^r]. 

55.  Ex.  I.  —  If  r„  r,,  are  the  p.  e.  of  the  measured  sides  AB,  BC,  of  a 
rectangle  ABCD,  find  the  p.  e.  of  the  area  of  the  rectangle. 

[Here  F  =  M,Af.. 

.-.   by  differentiation,         dF  =  M\dMo  +  M.jdM^, 
and  r^  =  M,^r}  +  J/,V,^] 

Ex.  2.  —  The  expansions  of  the  steel  and  zinc  bars  of  tube  i  of  the  Repsold 
base  apparatus  of  the  U.  S.  Lake  Survey  for  1°  Fahr.  are  approximately 

;«;;/.  vim. 

S  =  0.0248  J;  o.oooi. 
Z  =  0.0617  zk  0.0003. 

Show  that  ^  =  -  zt:  —  nearly. 

Z      5      400 

[For  ^=Z' 

.'.  dF  =  ^  dS  —  -y^  dZ., 

I  6"^ 

and  (p.  e.)2  =  ^  (o.oooi)^  +  —  (0.0003)^] 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         65 

Ex.  3.  —  The  base  b  and  the  adjacent  angles  ^,  C  of  a  triangle  ABC  are 
measured.  If  their  p.  e.  are  respectively  /-^,  ;'^,  r^,  tind  the  p.  e.  of  the 
angle  B  and  of  the  side  a. 

To  find  ;-^. 

We  have,  5  =  iSo  +  e  -  .4  -  C, 

where  e  denotes  the  spherical  excess  of  the  triangle. 
Hence,  A  and  C  being  independent  of  one  another, 

^B  =  ''a  +  ''(/• 
,  sin  A 
To  find  r„,  '^  =  ^iir^- 

By  differentiation, 

,        sin,-^    .,        ,  sin  (C  —  e)     .      ,,    ,  ^    ,  ^  r>    •       ,1   jr- 

da  =  -^ — r,  db  +  b ^-— n —  sin  i"  dA  +  a  cot  j9  sin  i"  c/C, 

sin  i?  sin-  B 

and  therefore, 

,      sin^^     ,  ,  (5- sin^  (C— e)  sin' i"      ,  ,     ,      ^2  e>    ■  ->    //      1 

r^  =    .   „  „  n'  + .    ■  J r }  -^  a}  cot*  5  sin-  i"  ;>'. 

«       sin^  ^   *  sin*  5  -^  *" 

Ex.  4.  —  Given  the  base  /;  and  the  angles  A^  B  oi  a.  triangle  with  p.  s.  e. 
r^.  r^,  r^,  respectively,  to  find  the  p.  e.  r^  of  the  side  a. 

ITT    ,  ,  sin  ^ 

W  e  have  a  =  b  -.     =  •  (i) 

sin  B 

This  might  be  expanded  as  in  the  preceding  example,  but  more  conve- 
niently as  follows  : 

Take  logarithms  of  both  members.     Then 

log  a  =  log  b  +  log  sin  A  —  log  sin  B.  (2) 

(a)    By  differentiation, 

da  =  ^  db  +  a  cot  A  sin  i"  dA  —  a  cot  B  sin  i"  dB. 

b 

Hence,  rj  =  ']-,  r^  +  a-  cot-  A  sin-  i"  r  /  +  ^-  cot-  B  sin=  i"  r^/.  (3) 

If,  as  is  usually  assumed  in  practice, 

r^^  =  fji^  r  and   ;-^  =  o, 

then                              r^^  =  a  sin  i"  r  \/cot=  ^  +  cot*  v9.  (4) 
(^>  Using  log  differences,  we  have  by  differentiating  (2), 

Bda  =  \db  +  S.jrt'^^  -  n,,dB,  (5) 

where  5  ,  5.  are  the  differences  corresponding  to  one  unit  for  the  numbers  a 


66  THE    ADJUSTMENT    OF    OBSERVATIONS 

and  b  in  a  table  of  logarithms,  and  5  ,,  5yj  are  the  differences  for  i"  for  the 
angles  A  and  B  in  a  table  of  log  sines.     Hence 

The  two  equations  (3)  and  (o)  may  be  used  to  check  one  another. 

The  above  formulas  are  true  only  when  the  angles  A  and  B 
are  absokitely  independent  of  each  other. 

This  caution  is  necessary  because  some  far-reaching  fallacies 
as  to  what  constitute  good  figures  in  triangulation  have  resulted 
from  overlooking  it. 

Ex.  5.  —  The  following  example  is  given  for  the  sake  of  showing  the  form 
of  solution  by  the  method  of  logarithmic  differences. 

In  the  triangulation  of  Lake  Superior  there  were  measured  in  the  triangle 
Middle,  Crebassa,  Traverse  Id   {ABC), 

Z^  =57°  04' 51.4"     r,  =  0.30", 
AB  =  (^t  15'  39--"     rj,  =  0.29". 

The  side  Middle-Traverse  Id.  as  computed  from  the  Keweenaw  Base  is 
16894.9  yards.     Taking  r^  =  0.05  yd.,  find  7\^  and  7■^^. 

We  have  a  =  i>  —. — f,* 

sin  B 

.:  log  (a  +  da)  =  log  (b  +  db)  +  log  sin  [A  +  dA)  -  log  sin  {B  +  dB). 

Then,  the  differences  being   expressed    in  units   of   the   seventh   decimal 
place, 

log  ib  +  db)  =  4.2277556  +  257  db 

log  sin  {A  +  dA)     =  9.9239892  +    14  dA 

colog  sin  {B  +  dB)  =  0.0351398  —      9  dB 

:.  log  a  +  5^^da        =  4.1868846  +  257  db  +  14  dA  —  c)dB, 
and  283  da  =257  db  +  14  dA  —  9  dB, 

since  log  a  =  4.1868846, 

and  283  is  the  difference  5^  as  given  in  the  table. 

Hence  ,-.  (:|J(.05;=  +  (±LJ(.30,'  +  (^-J(.9)' 


and 

=  0.0024, 
r„  =  0.05  yd. 

Also, 

=  \/(0.29)2  +   (0.30)2 
=  0.42". 

OBSERVATIOxXS    OF    OXE    UXKXOWN    QUAXTITY         67 

Miscellaneous  Examples. 
56.  Examples  of  Probable  Error. 

Ex.  I.  —  If  in  a  theodolite  read  by  2  verniers  the  p.  e.  of  a  reading  (mean  of 
vernier  readings)  is  2",  show  that  if  it  is  read  by  3  verniers  the  p.  e.  of  a 
reading  will  be  a  little  over  1.5",  and  if  read  by  4  verniers  a  little  less  than 
1.5".  provided  the  errors  are  of  the  accidental  class. 

Ex.  2.  —  The  p.  e.  of  an  angle  of  a  triangle  is  /- ;  show  tliat  the  p.  e.  of  the 
closing  error  of  the  triangle  is  ;-  \'3,  all  of  the  angles  being  equally  well 
measured. 

[Closing  error  =  180°  -  {A  ^  B  ^-  C).\ 

Ex.  3.  —  The  length  of  a  measuring  bar  at  the  beginning  of  a  measurement 
was  a  -^  ;-,.  After  x  measures  had  been  made,  it  was  b  ^  r.,.  Show  that 
the  length  of  the  ii\\\  measure,  the  length  being  supposed  to  change  uniformly 
with  the  distance  measured,  is 


[For  if  da  is  the  error  of  a,  and  db  of  /'.  then  the  error  of 

a  ^  -  [b  -  a)    is    (  i  -  'M  ^/a  +  "  db, 
X  \  x)  X 

and  the  above  p.  e.  follows. 

It  is  a  common  mistake  to  write  the  error  in  the  form  da  +  ~  {db  —  da), 

/  n'^  ^ 

and  hence  to  infer  that  the  p.  e.  is  i  /  r^  H — ,  {r^  +  r^).  ] 

Ex.  4.  —  Prove  that  the  p.  e.  of  the  mean  of  two  observations  whose  dif- 
ference is  d\s  0.337  d,  and  the  p.  e.  of  each  observation  is  0.477  d. 

Ex.  5.  —  The  line  Monadnock-Gunstock  (94469  m.)  was  computed  from  the 
Massachu.setts  Base  (17326  m.)  through  the  intervening  triangulation.  The 
p.  e.  of  the  line  arising  from  the  triangulation  is  -|-  0.317  m.,  and  the  p.  e.  of 
the  base  is  0.0358  m. ;  find  the  total  p.  e.  of  the  line. 


P-  ^-  =  V  V^ToS  ^  °-°^58J'+  (0.317)-  =  i  0.372  m. 
Ex.  6.  —  The- Minnesota  Point  Base  reduced  to  .sea-level  is 
1325  X  15  ft.  bar  at  32°  +  11.314  in.  J^  0.421  in. 
and  15  ft.  bar  at  32°  =  179.9543S  in.  ^^i^  0.00012  in. 

show  that  the  p.  e.  of  the  base  is  ^  0.450  in. 

[p.  e.  =  ^(1325  X  0.00012)^  +  (0.421)' =  J;;0.45oin.  We  multiply  J- 0.00021 
inches  by  1325  :  uncertain  which  sign  it  is;  but  whichever  it  is,  it  is, constant 
all  the  way  through.] 


68  THE   ADJUSTMENT    OF    OBSERVATIONS 

Ex.  7.  —  If  the  zenith  distance  f  of  a  star  is  observed  n^  times  at  upper 
cuhnination,  and  the  zenith  distance  f '  of  the  same  star  is  observed  n^  times 
at  lower  culmination,  show  that  the  m.  s.  e.  of  the  latitude  of  the  place  of 
observation  is 


U         /  I  I 

2  y  «i    «2 


M  being  the  m.  s.  e.  of  a  single  observation. 

[Latitude  =  90°  -  i  (f  +  fO-] 
Ex.  8.  —  Given  the  telegraphic  longitude  results, 

h.     ;«.        J.  s, 

Cambridge  west  of  Greenwich  =  4  44  30.99  -j-  0.23 

Omaha,  west  of  Cambridge        =  i  39  15.04  J-  o  06 

Springfield  east  of  Omaha  =      25  08.69^0.11 

show  that     Springfield  west  of  Greenwich  =  5  5S  37.34  -^  0.26 


[p.  e.  =  V.23-  +  .06-  +  .11-  =  0.26.] 


Ex.  g.  —  Given  mass  of  earth  +  mass  of  moon  = , , 

305,879^2271 

and  mass  of  moon  = mass  of  earth, 

i>i.44 

prove  mass  of  earth  = 


3091635  ±  2299 
For  (305,879  i  2271)  X  g^  =  309,635  ±  2299. 
Ex.  ID.  —  In  measuring  an  angle  suppose 

7\  =  p.  e.  of  a  pointing  at  a  signal, 
Tj  =  p.  e.  of  a  reading  of  the  limb  of  the  instrument, 
e  =  error  of  graduation  of  the  arc  read  on  ; 

then,  assuming  tliat  these  result  from  the  only  sources  of  error  not  eliminated, 
show  if  the  limb  has  been  changed  ;;/  times,  and  ;/  readings  taken  in  each 
position,  that 

p.  e.  of  angle  =  J^  i  / 

For  one  position  of  the  limb, 

p.  e.  of  angle  =  i  1/ 

Y  /t 

as  the  error  of  graduation  remains  constant  throughout  each  set  of  n  read- 
ings. 

It  is  important  to  note  that  the  ;/-i  readings  in  each  position  after  the 
first  reading  has  been  taken  reduce  the  effects  of  pointing  and  reading 
errors  but  not  of  errors  of  graduation. 


h- 

(r 

^+r 

3=) 

+ 

e" 

1 

mn 

VI 

I- 

(r_ 

^  +  r 

/) 

+ 

<?-, 

OBSERVATIONS    OF    OXE    UNKNOWN    QUANTITY  69 

What  would  the  p.  e.  of  an  angle  have  been  if  each  of  the  ;//;/  readings 
had  been  taken  in  a  new  position  ? 

Ex.  II.  —  Thedistanceo  —  i  mm.  on  a  graduated  line-measure  is  read  with 
a  micrometer ;  show  that  the  p.  e.  of  the  mean  of  two  results  is  equal  to  the 
p.  e.  of  a  single  reading. 

[For  distance  o  —  i  mm. 

=  i  {(first  +  second  rdg.)  at  o  -  (first  +  second  rdg.)  at  i  mm.} 
•••  (P-  e.)=  =  i{4  (P-  e.)2  of  a  rdg.}] 

Ex.  12.  —  In  the  comparison  of  a  mm.  space  on  two  standards  placed  side 
by  side  and  read  with  a  micrometer,  the  p.  e.  of  a  single  micrometer  reading 
being  a,  show  that  the  p.  e.  of  the  difference  of  the  results  of  n  combined 

measurements  (each  being  the  mean  of  two  measurements)  is  1/ -«. 
[For  p.  e.  of  a  reading  =  a. 

.•.  p.  e.  of  a  combined  measurement  =  a, 
and  p.  e.  of  mean  of  n  combined  measurements  =  — -  <  etc.] 

\    71 

Ex.  13.  —  A  theodolite  is  furnished  with  11  reading  microscopes,  all  of  the 
same  precision.  A  graduation-mark  on  the  limb  is  read  on  ///  times  with  a 
single  microscope,  giving  the  p.  e.  of  a  single  reading  to  be  7\.  The  telescope 
is  then  pointed  at  an  object  m  times,  and  the  p.  e.  of  the  mean  of  the  micro- 
scope readings  is  found  to  be  7\.     Show  that  the  p.  e.  of  a  pointing  is 


\J' 


p.  e.  of  reading  (mean  of  verniers)  with  ;/  microscopes  =  — L.. 

L  v« 

Total  error  =  error  of  reading  -f  error  of  pointing, 
.'.r/  =  -'-  +  (p.  e.  of  ptg.)^  etc. 

Ex.  14.  —  If  ;'i/^,,  r.Jh,  are  the  p.  e.  of  the  base  measurements,  and  ;-.,X  the 
p.  e.  of  the  ratio  \,  given  by  the  triangulation,  of  a  base  A,  to  a  base  /^,, 
show  that  the  p.  e.  of  the  discrepancy  between  the  computed  and  measured 
values  of  b.,  is  b.,  V[r'-]. 

[Discrepancy  =  b.,  —  b\  =  /suppose. 
.*.    d!>;  —  b^d\  —  \dbi  =  dly 
and  r^^.,^  +  b,\r.,^^^'  +  \\r,b,f  =  r", 

or  bi  (r/  -f  r.}  +  r,^)  =  r^.] 

Ex.  15.  —  At  the  time  /,  the  correction  to  a  chronometer  was  a'  ^  /•,,  and 
at  the  time /.Jt  was  a/ J;  ;-., ;  show  that  tlic  p.  e.  of  the  rate  of  the  cliro- 

\/;'  2  _|-  ^  2 

nometer  is and  find  the  p.  e.  of  the  correction  to  tlir  chronometer 

*2  ~~  '1 

at  an  interpolated  time  t'. 


70  THE    ADJUSTMENT    OF    OBSERVATIONS 

Correction  =  a^  +  -j j-if'  —  ty)  at  time  f. 

V(4  -  nW  +  if  -  tx?r.{ 


Ex.  i6,  —  Given  x  cos  a  =  /,  J^  r^ 

X  sin  a  =  /o  zt  ^2> 


■ 


find  p.  e.  of  X  and  of  a. 


,/.,.  ^  ^  ,^/^  +  ^^  ,^4 


^-   ,^/;+  _A^^/ 


.-.  p.  e.  oix=U      1^^  _^  ^-3      •    Similarly  p.  e.  of  a  =  — -  ^,^,  '    ^- 

Ex.  17.  —  Given  on  a  line-measure  the  p.  e.  of  a  distance  OA  measured 
from  O  to  be  7\,  and  of  OB,  also  measured  from  O,  to  be  r, ;  find  the  p.  e. 
of  OD  when  D  is  the  middle  point  of  AB. 

[OB  =  h{OA  +  OB). 

If  Tj  =  ^2  =  i^,  for  example,  then  r  =  .08'*,  when  fi  =  one  micron. 

It  may  at  first  sight  appear  paradoxical  that  the  p.  e.  of  the  computed 
quantity  may  be  smaller  than  the  p.  e.  of  the  measured.  It  is  evident,  how- 
ever, that  the  error  of  OD  is  one-half  the  sum  of  the  errors  of  OA  and  OR. 
If  the  signs  of  the  errors  are  alike,  the  error  of  OD  is  never  greater  than  the 
larger  of  the  errors  ;  if  the  signs  are  different,  it  is  always  less.] 

Ex.  18.  —  Given  the  p.  e.  of  x  to  be  r ;  find  the  p.  e.  of  log  x. 

I  d  log  X  = '-  dx. 


p.  e.  log  X 


mod.       I 

-  =  r. 

X  J 


Ex.  19.  —  In  the  measurement  of  the  Massachusetts  base  line,  consisting  of 
2165  boxes,  the  p.  e.  of  a  box,  as  derived  from  comparisons  with  the  standard 
meter,  was  j-  0.0000055  m.,  the  p.  e.  from  instability  of  microscopes  in  mea- 
suring a  box  was  0.000127  m.,  and  the  p.  e.  of  the  base  from  temperature  cor- 
rections was  0.0332  m.  Show  that  the  p.  e.  of  the  base  arising  from  these 
independent  causes  combined  is  0.0358  m. 

[p.  e.  =   v(2i65  X  0.0000055  m.)^  +  (0.000127  m.  ■\/2i65)^  -f  (0.0332  m.)' 
=  ±00358  m.] 

Ex.  20.  —  Given  the  length  of  the  Massachusetts  base  to  be  17326.3763  m. 
j;  0.0358  m.;  show  that  the  corresponding  value  of  the  p.  e.  of  its  logarithm 
is  8.973  in  units  of  the  seventh  place  of  decimals. 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         71 

[log  [b  ±  0.0358)  =  log  ^  ±  "^  (0.0358). 

log  mod.    9.6377843 
log  0.0358  8.5538830 

8.1916673 
log  b  4.2387077 


0.0000008973  3.9529596  ] 

Ex.  21.  —  The  p.  e.  of  the  log  of  a  number  N  in  units  of  the  seventh 
decimal  place  is  ^  10.6 ;  find  the  ratio  of  the  p.  e.  to  the  number. 


/mod.\2 
mod 


r-\og{N->rv) 


\r\ 


N 


r  =  10.6  -^  10', 


and  ^,  =  — i— . 

yv     410,000  J 

57.    Examples  of  the  Weighted  Mean. 

Ex.  I.  —  The  weights  of  the  independently  measured  angles  BAC,  CAD, 
DAE,  are  3,  3,  i  respectively  ;  find  weight  of  the  sum-angle  BAE.    Ans.  oX- 

To  solve  this,  write  the  expression  for  (p.  e.)^  In  general,  the  weight  of 
the  combination  of  several  quantities  united  by  +  or  —  signs,  thus, 

X=Xa-X,-  X.+    ... 

is  given  by  the  expression 

-r  pa         pb         pc 

Ex.  2.  —  If  X  —  a,Xj  +  a.,x.,  +  .  .  .  +  <?n;rn,  and  /,,  p.,,  .  .  .  p„  are  the 
weights  of  Xj,x.,,  .  .  .  x„,  and  /"  the  weight  of  X,  show  that 


I   _  Vaa] 


P 

Ex.  3.  —  Professor  Hall  found,  from  observations  of  the  satellites  of  Mars, 

that  from    Deimos,  mass   of   Mars  = r — rr  '  and  from  Phobos, 

3,o95»3i3  ±  34«5 

mass  of  Mars  = -r — 7—, ,  the  mass  being  expressed  in  the  common 

3,078.456-1-10,104 

unit.     Show  that,  taking  the  weighted  mean,  we  have  approximately, 

mass  of  Mars  = j 

3,093,500  i  3295 

Ex.  4.  —  On  a  graduated  bar  the  space  o  —  i  m.  is  measured  and  found 


7i  THE    ADJUSTMENT    OF    OBSERVATIONS 

to  be  I  m.  with  a  weight  i,  and  the  space  o  —2  m.  is  measured  and  found  to 
be  2  m.  with  a  weight  2  ;  required  the  vahie  of  the  space  i  m.  —  2  m.  and  its 
weight  P. 

[Space  I  m.  —  2  m.  =  i  m.     It  makes  no  difference  what  the  weights  are 
so  far  as  the  value  of  the  space  is  concerned. 
To  find  /•,  (i  m.  —  2  m.)  =  (o  —  2  m.)  — (o  —  i  m.). 


=  3] 


.-.  ^  =  -  +  -  =  ^  and  Z'  = 

Ex.  5.  —  Given  the  weight  oi  x  =  p.,  show  that 

weight  of  log  ^  =  ^^,  A 

Ex.  6. —  If  X  =  -  and  the  weight  oi y  \%  p,  then 

weight  of  X  =  c'^p. 

Ex.  7.  —  Given  the  results  for  difference  of  longitude,  Washington  and 
Key  West, 

m.         s-  s. 

1873,  Dec.  24,  19  01.42 -|- 0.044 
Dec.  26,  1.37  i  .037 
Dec.  30,  1-38  i  -036 
Dec  31,  1.45  ±    -036 

1874,  Jan.     9,  1.60^    .046 


Jan.  10,  1-55  rt    -045 

Jan.  II,  19  01.57  jt:  0.047 


show  that 


Weighted  mean  =  19  01.460^0.016, 

Weighted  mean  of  first  four  nights  =  19  01.404 -|- 0.019, 
Weighted  mean  of  last  three  nights  =  19  01.573  j-  0.027, 

and  from  the  last  two  results  check  the  first. 

Ex.  8.  —  In  the  triangulation  connecting  the  Kent  Id.  Base,  Md.,  and  the 
Craney  Id.  Base,  Va.,  the  length  of  the  line  of  junction  computed  from 

tn.  711. 

Kent  Id.  Base       =  26758.432  -1-  0.3S, 
Craney  Id.  Base  =  26758.176  ^^^  0.43. 
Show  that 

}n.  in. 

(i)  Discrepancy  of  computed  values    =  0.2564:0.57. 

(2)  Most  prob.  length  of  junction  line  =  26758.32    4;  0.28. 
Ex.  9.  —  In  latitude  work  with  the  zenith  telescope,  if  n  north  stars  are 
combined  with  s  south  stars,  giving  ns  pairs,  to  find  the  weight  of  the  com- 
bination, that  of  an  ordinary  pair,  one  north  and  one  south,  being  unity. 

[Let  r  =  the  p.  e.  of  an  observation  of  one  north  star  or  of  one  south 
star. 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         73 

Then,  as  though  combining  the  mean  of  //  north  stars  with  the  mean  of 
s  south  stars,  the  \vt.  /  of  the  combination  is 

I 

P 
But  I 

"The  combination  of  more  than  two  stars  gave  some  trouble.  In  one 
case  there  were  3  north  and  4  south,  which  would  give  12  pairs,  but  with  a 

weight  of  2- ^only.     In  this  and  all  similar  cases  I  treated  the  whole 

3+4 
combination  as  one  pair;  that  is,  I  inserted  in  the  blank  provided  the  half- 
sum  of  the  mean  of  the  declinations  of  north  stars  and  of  the  mean  of  the 
declinations  of  south  stars,  and  gave  the  result  a  higher  weight.  This  is  the 
only  logical  method."  (Safford,  Report^  Chief  of  Engineers  U.  S.  A.,  1S79, 
p.  1987.) 

For  a  series  of  examples  by  Airy  on  the  weights  to  be  given  to  the  sepa- 
rate results  for  terrestrial  longitude  determined  by  the  observations  of 
transits  of  the  moon  and  fixed  stars,  see  A/e///.  Koy.  Astron.  Soc,  vol.  xix. 

Ex.  10.  —  If  a  close  zenith  star  is  observed  with  a  zenith  telescope  first  as 
a  north  star,  and  immediately  after  as  a  south  star,  show  that  the  weight  of 
the  resulting  latitude  is  less  than  that  found  from  observing  an  ordinary  pair. 

Ex.  II.  —  In  the  triangulation  of  Lake  Ontario  the  angle  Walworth-Pal- 
myra-Sodus  was  measured  as  follows: 

In  1875,  with  theodolite  P.  and  M.  No.  i, 

74°  25'  05.429",  J;;  0.29",  mean  of  16  results. 

In  1877,  with  theodolite  T.  and  S.  No.  3, 

74°  25'  04.611"^  0.22",  mean  of  24  results  ; 

required  the  most  probable  value  of  the  angle  and  its  probable  error 

•    (0.22)'' 

The  weights  are  in  the  ratio r^- 

^  (0-29)^ 

Note. —  If,  instead  of  being  two  measurements  of  the  same  angle,  the 
above  were  the  measurements  of  two  angles  side  by  side,  then 

total  angle  =  148°  50'  10.040", 

because,  no  matter  how  much  better  one  is  measured  than  the  other,  we  can 
do  nothing  but  take  the  sum  of  the  two  values. 

Ex.  12.  —  An  angle  is  measured  /i  times  with  a  repeating  theodolite,  and 
also  n  times  with  a  non-repeating  theodolite,  the  precision  of  a  single  reading 
and  of  a  single  pointing  being  the  same  in  both  cases ;  compare  the  weights 
of  the  results. 


74  THE     ADJUSTMENT    OP    OBSERVATIONS 

[r,,  rjthe  p.  e.  of  a  single  pointing  and  of  a  single  reading. 

With  a  non-repeating  theodolite  each  measurement  of  the  angle  contains 

(pointing  +  reading)  —  (pointing  +  reading) 
.•.  (p.  e.)^  of  one  measurement  =  z  r^  -V  z  r.?, 

and  (p-  e.)^  of  mean  of  n  measurements  =     -  (2  r?  +  2  r.?). 

n 

With  a  repeating  theodolite  the  successive  measurements  of  the  angle  are 

(pointing  +  reading)  —  pointing 
pointing  —  pointing 


pointing  —  (pointing  +  reading). 
.•.  (p.  e.)^  of  ft  times  the  angle  =  2  nr^  +  2  r^, 

and  (p.  e.)'  of  the  angle  =  —  (2  ?ir-^+  2  r^). 

If,  then,  /i,  p..  denote  the  weights  of  an  angle  resulting  from  n  reiterations 
or  from  n  repetitions, 

and  hence  it  would  seem  that  the  method  of  repetition  is  to  be  preferred  to 

f-i 
the  method  of  reiteration.     This  advantage  is  so  much  less,  the  smaller  -^ 

is ;  that  is,  the  more  the  precision  of  the  circle  reading  increases  in  propor- 
tion to  the  precision  of  the  pointing. 

This  result  is  contradicted  by  experience. 

The  result  obtained  is  true  on  the  hypothesis  that  only  accidental  errors 
enter.  We  have  assumed  a  perfect  instrument.  But  the  instrument-maker 
cannot  give  what  the  geometer  demands.  From  various  mechanical  reasons 
the  systematic  error  in  a  repeatmg  theodolite  increases  with  the  number  of 
observations,  whereas  in  the  reiterating  theodolite  it  disappears.  This  sys- 
tematic error,  in  whatever  way  it  arises,  causes  the  trouble.  It  is  so  difficult 
to  eliminate  the  systematic  error  in  observations  with  a  repeating  theodolite, 
that  in  spite  of  its  advantages  over  the  direction  theodolite  in  so  far  as  acci- 
dental errors  are  concerned,  it  is  still  an  open  question  which  type  of  instru- 
ment will  give  greater  accuracy.  In  the  Coast  and  Geodetic  Survey  both 
types  are  in  use.  The  direction  theodolite  is,  however,  used  much  more  than 
the  repeater  on  primary  triangulation. 

Of  tJie  WeigJiting  of  Ohsejvations. 

58.  When  the  sources  of  error  are  of  such  kinds  that,  so  far 
as  we  know,  they  cannot  be  separated,  the  p.  e.  and  consequent 
weight  are  found  as  described  in  the  preceding  sections.  The 
weight  has  been  defined  as  a  number  representing  the  relative 


OBSERVATIONS    OF    OXE    UNKNOWN    QUANTITY         75 

goodness  of  an  obsen^ation,  or  of  a  result  computed  from  obser- 
vations, and  as  a  number  inversely  proportional  to  the  square  of 
the  p.  e.  of  an  observation  or  computed  result.  In  assigning  or 
in  computing  weights  it  must  be  kept  continually  in  mind  that 
the  intention  is  to  make  weights  inversely  proportional  to  the 
squares  of  the  p.  e.  arising  from  all  sources,  and  that  no  avenue 
of  information  should  be  neglected.  To  assign  weights  which 
are  simply  inversely  proportional  to  the  computed  p.  e.'s  which 
may  happen  to  be  available  without  careful  examination  as  to 
their  trustworthiness  is  not  good  practice,  nor  is  it  in  accordance 
with  the  complete  theory  of  the  method  of  least  squares.  The 
object  of  the  following  paragraphs  is  to  indicate  some  of  the 
precautions  which  must  be  taken  in  assigning  weights. 

For  example,  two  separate  determinations  of  a  millimeter 
space,  made  in  the  same  way,  gave 

1 000. 1  ±  0.40,  mean  of  20  readings, 
1000.3  i  °*33'  iiiean  of  30  readings. 

To  find  the  weighted  mean  of  these  two  sets  of  measurements, 
we  may  proceed  in  two  ways.  The  number  of  results  in  the 
first  measurement  is  20,  and  the  number  in  the  second  is  30. 
Hence,  taking  the  weights  proportional  to  the  number  of  results, 
the  mean 

20  X  0.1  +  30  X  -3 

=  1000  -\ =  1000.22. 

20  +  30 

Again,  since  the  p.  e.  of  the  measurements  are  0.40 and  0.33, 

their  weights  are  as        ,  to  — ,,   that  is,  as    1089  to   1600,  and 

40-        33" 
the  resulting  weighted  mean  is  1000.22,  agreeing  with  the  other 
computed  value  to  the  last  decimal  place  retained. 

59.  In  this  example  the  two  methods  of  computation  give 
exactly  the  same  result.  It  is  not  always  so.  Some  "  run  of 
luck,"  or  balancing  of  errors,  or  constant  conditions,  might  have 
made  the  observations  of  one  set  fall  very  closely  togetluT,  in 
which  case  the  weigiit  as  computed  from  the  p.  c.  would  have 


76  THE    ADJUSTMENT    OF    OBSERVATIONS 

been  very  large,  while  in  the  other  set  varying  conditions  might 
have  caused  large  ranges  and  the  computed  weight  would  have 
been  small.  In  such  a  case  the  two  means  might  differ  consider- 
ably. It  would  then  be  desirable  to  study  carefully  the  question 
as  to  which  method  of  weighting  is  probably  the  better.  In 
such  a  study  it  must  be  kept  in  mind  that  the  computed  probable 
errors  derived  from  the  observations  are  subject  to  uncertainty 
for  that  reason,  and  are  peculiarly  subject  to  being  largely 
increased  by  one  or  a  few  large  residuals.  If  two  series  of  ob- 
servations made  under  similar  conditions  by  the  same  observer 
gave  computed  probable  errors  which  differed  widely,  it  would 
be  an  open  question  whether  the  actual  p.  e.  of  the  two  series 
differed  widely  or  whether  the  difference  was  simply  apparent 
and  due  to  each  of  the  computed  p.  e.'s  being  based  on  too 
small  a  number  of  observations  to  be  a  close  approximation  to 
the  truth.  The  computer  must  use  his  judgment  in  deciding 
this  question  and  assign  the  weights  accordingly.  Thus,  in  the 
second  set  of  observations  above  the  first  three  results  were  999.8, 
999.8,  999.8.  The  p.  e.  computed  from  these  values  would  be 
zero  and  the  consequent  weight  infinite.  But  no  one  will  doubt 
that  these  observations  are  subject  to  some  error,  and  that  the 
weights  assigned  to  them  should  be  finite  and  smaller  than  the 
weight  assigned  to  the  mean  of  the  thirty  observations. 

60.  An  Approximate  Method  of  Weighting.  — A  long-con- 
tinued series  of  observations  will  show  the  kind  of  work  an 
instrument  is  capable  of  doing  under  favorable  conditions  ;  and 
if  work  is  done  only  when  the  conditions  are  favorable,  the  p.  e. 
derived  from  a  certain  number  of  results  will  generally  fall  within 
limits  that  can  be  assigned  a  priori.  For  example,  with  the 
Lake  Survey  primary  theodolites,  which  read  to  single  seconds,  the 
tenths  being  estimated,  the  work  of  several  seasons  showed  that 
the  p.  e.  of  the  mean  of  from  16  to  20  results  of  the  value  of  a 
horizontal  angle,  each  result  being  the  mean  of  a  reading  with 
telescope  direct  and  of  a  reading  with  telescope  reverse,  need  not 
be  expected  to  be  greater  than  0.3".     If,  therefore,  after  having 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         77 

measured  a  series  of  angles  in  a  triangulation  net  with  these 
instruments,  the  p.  e.  all  fell  within  ±  0.3",  it  was  considered 
sufficiently  accurate  to  assign  to  each  angle  the  same  weight. 

The  objection  to  this  is  that  "an  instrument  which  has  a 
large  periodic  error  may,  if  properly  used,  give  as  good  results 
as  if  it  had  none  ;  but  the  discrepancies  between  its  combined 
results  for  an  angle  and  their  mean  ma}-  be  large,  thus  giving  an 
apparently  large  probable  error  to  the  mean.  Moreover,  a  given 
number  of  results  over  short  lines,  or  lines  over  which  the  dis- 
tant signals  are  habitually  steady  when  seen  in  the  telescope, 
will  give  a  resulting  value  for  the  angle  of  much  greater  weight 
than  the  same  number  of  combined  results  between  two  stations 
which  are  habitually  unsteady."  * 

The  same  method  of  weighting  was  employed  by  the  Northern 
Boundary  Commission  in  their  latitude  work.  *'  The  standard 
number  of  observations  [for  a  latitude  determination]  was  finally 
fi.xed  at  about  60,  it  being  found  that  with  the  32-in.  instrument 
60  observations  would  giv^e  a  mean  result  of  which  the  p.  e. 
would  be  about  4  feet."f  This  method  of  weighting  is  based 
upon  the  idea  that  a  closer  approximation  to  the  truth  is  obtained 
in  these  cases  by  assuming  that  the  actual  p.  e.'s  are  all  equal 
than  by  accepting  their  separate  computed  values  as  represent- 
ing the  facts  accurately.  In  other  words,  such  a  procedure  is 
equivalent  to  assuming  that  the  principal  cause  of  variation  of 
the  separate  computed  p.  e.  from  a  mean  value  is  the  accidental 
grouping  of  unusually  large  or  unusually  small  residuals  rather 
than  a  real  variation  in  accuracy  between  the  different  series. 

61.  Weighting  when  Constant  Error  is  Present. — The 
preceding  leads  us  to  the  case  where  the  error  of  observatioji 
can  be  separated  into  two  parts,  one  of  which  is  due  to  acciden- 
tal causes,  and  the  other  to  causes  which  are  constant  through- 
out the  observations.  The  total  error  e  would,  therefore,  be  of 
the  form  e  =  /,  +  /,. 

*  Professional  Papers  of  the  Corps  of  Engineers  U.  S.  A.,  No.  24,  p.  354. 
^  Report.,  Survey  of  the  Northern  Boundary,  p.  86. 


78  THE    ADJUSTMENT    OF    OBSERVATIONS 

This  case  has  been  discussed  already  in  general  terms  in  Art.  40 
in  explaining  the  well-known  fact  that  an  increase  in  the  number 
of  observations  with  a  given  instrument  does  not  lead  to  a  cor- 
responding increase  of  accuracy  in  the  result  obtained. 
Let 

r  =  the  p.  e.  of  the  observation  arising  from  the  accidental 

causes, 
r^  =  the  error  peculiar  to  the  observation  arising  from  the 
constant  causes. 
Then  r^,  r^,  being  independent,  and  being  as  likely  to  have 
opposite  signs  as  the  same  sign,  the  total  p.  e.,  r  of  observation 
may  be  assumed  (Art.  53), 

;-2   =   ;-^2    +   ,,2^ 

If  n  observations  have  been  made,  we  shall  have  for  the  p.  e.  r^ 
of  their  mean,  since  r^  is  constant, 

y2 

'0      T^    '2 

n 
It  is  evident  that  when  n  is  large,  n"  becomes  the  important 
term,  and  that  in  any  case  the  value  of  r^  and  consequent  weight 
can  be  but  little  improved  by  increasing  the  number  of  observa- 
tions. 

62.  For  the  purpose  of  finding  the  value  of  the  p.  e.  arising 
from  the  constant  sources  of  error,  a  special  series  of  observa- 
tions is,  in  general,  necessary.  After  this  series  has  been  made, 
the  value  of  r.  found  from  it  can  be  applied  in  the  determination 
of  the  value  of  r^  in  any  other  series  made  under  like  conditions. 

For  illustration  let  us  consider  a  latitude  determination  with 
the  zenith  telescope.  The  zenith-distance,  ^,  of  each  star  being 
observed,  the  half -difference  of  zenith-distances  for  each  pair 
may  be  computed,  and  each  of  these  computed  values  may  be 
considered  an  observed  value.  The  values  of  the  declinations  h 
are  taken  from  a  catalogue  of  stars.  The  errors  of  S  are,  there- 
fore, independent  of  those  of  K,  and  are  constant  for  the  same 
pair  of  stars.     The  latitude  </)  from  one  pair  is  given  by 


Let 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY        79 

(^  =  i  (8'  +  8)  +  Kr  -  0- 

r^  =  the  p.  e.  of  +  (C  —  0  for  one  observation  of  one  pair, 

rs  =  the  p.  e.  of  4  (8'  +  8)  for  this  pair, 

rA,  =  the  p.  e.  of  the  resulting  latitude  <^  from  one  pair, 

then  for  a  single  observation  of  this  pair, 

r^^  =  ''5^  +  r^, 

and  for  u  observations  of  this  pair, 

^  n 

The  quantity  r^  will  be  found  from  repeated  observations  of 
the  same  pair  of  stars,  as  the  error  in  declination  will  not  influence 
the  result.  A  better  value  will,  of  course,  be  obtained  from 
several  pairs  than  from  a  single  pair.  Let,  then,  many  pairs  of 
stars  be  observed  night  after  night  for  a  considerable  period. 
Collect  into  groups  the  latitudes  resulting  from  the  obser\'cd 
values  of  each  separate  pair.  Let  «,,  w^,  .  .  .  «,„  be  the  number 
of  results  in  the  several  groups,  the  number  in  any  group  being 
at  least  two.  Form  the  residuals  for  each  group  and  compute 
the  p.  e.  in  the  usual  way.     We  have  : 


No.  OF 

Night. 

First  Pair. 

Second  Pair. 

Results. 

V 

Results. 

V 

.   .   . 

I 
2 

3 

0,' 

7'i' 

0/ 

<t>'' 

V.' 

. 

Means 

d> 

<p 

Now,  assuming  that  the  p.  c.  of  observation  of  each  pair  is  the 
same, 


So        THE  ADJUSTMENT  OF  OBSERVATIONS 


n^  —  I 


If,  then,  ;/  is  the  total  number  of  results,  and  in  the  number  of 
groups,  by  adding  the  above  equations  there  results 

_     W 


n  —  m 


In  finding  r^  we  assume  that  though  errors  of  declination  are 
constant  for  each  star,  still  for  a  latitude  found  from  many  pairs 
in  the  same  catalogue  these  errors  may  be  regarded  as  acciden- 
tal. Let,  then,  many  different  pairs  of  stars  be  observed  on 
each  of  n  nights  at  in'  places,  no  star  being  observed  at  more 
than  one  place.  Collect  the  means  of  the  single  results  of  each 
separate  pair,  and  form  the  residuals  z>'  for  each  place,  taking  the 
differences  between  these  means  considered  as  single  results  and 
their  mean  for  that  place.  Then,  reasoning  as  above,  the  p.  e. 
of  a  latitude  resulting  from  ji  observations  on  a  single  pair  of 
stars  is 


n' 


where  ii'  is  the  number  of  different  pairs  of  stars  observed,  and 
;//  is  the  number  of  places  occupied. 
Now,  /'s  is  found  from 


n 


and  is,  therefore,  known  for  the  star  catalogue  used.     This  value 
may  be  taken  in  future  work  in  finding  r^  from 

r^  =  ri  +  —  5 
n 

and  the  consequent  combining  weight  of  ^  will  be  as 

I 


OBSERVATIOXS    OF    ONE    UXKXOWX    OUAXTITY         8i 

63.  An  example  of  a  similar  kind  is  afforded  in  finding  the 
weights  of  the  angles  measured  with  a  theodolite  in  a  triangula- 
tion  where  more  rigid  values  are  required  than  would  be  found 
by  Art.  60.  The  actual  error  of  a  measured  value  of  an  angle 
arises  from  two  main  sources,  errors  of  graduation  and  errors  of 
observation.  The  former  are  constant  for  each  part  of  the 
limb  read  on,  and  correspond  to  the  declination  errors  above, 
while  the  latter  are  incapable  of  classification,  and  are,  therefore, 
assumed  to  be  accidental.  The  periodic  errors  of  graduation  are 
supposed  to  have  been  eliminated  by  proper  shiftings  of  the 
circle.  The  resultant  p.  e.  r  of  a  single  measurement  is  found 
from 

and  the  p.  e.  r^  of  the  mean  of  n  measurements  made  on  the 
same  part  of  the  limb  from 

where  r^,  r,  are  the  p.  e.  of  graduation  and  observation  respec- 
tively. The  method  of  treating  this  problem  is  quite  similar  to 
that  of  the  preceding :  ;;  is  found  by  reading  the  same  gradua- 
tion-mark on  the  limb  many  times,  and  ;;,  by  reading  the  angle 
between  two  fixed  signals  many  times,  the  limb  being  changed 
after  each  reading.  Thence  i\  is  known  for  the  instrument  in 
question,  and  the  combining  weights  of  angles  measured  with 
this  instrument  are  at  once  found. 

64.  The  foregoing  leads  to  another  important  practical  [loint 
in  the  measurement  of  angles.  If  the  weight  of  a  single  obser- 
vation is  unity,  then  the  weight  of  the  mean  of  ;/  observations 
made  with  the  limb  in  one  position  is 

^       r^^  +  rl 
n 
Yor  certain  instruments,  experience  has  shown  that  we  may 
safely  assume 


82  THE    ADJUSTMENT    OF    OBSERVATIONS 

and  therefore  it  follows  that  for  these  instruments 

2  n 
^       »  +  I 
Hence,  in  using  these  instruments,  no  matter  how  many  obser- 
vations we  make  in  one  position  of  the  limb,  we  never  reach  the 
precision  of  the  mean  of  two  observations  made  with  the  limb  in 
different  positions. 

It  is  evident  that  to  secure  the  maximum  efficiency  in  the 
elimination  of  error,  the  limb  of  a  direction  instrument  should  be 
shifted  after  each  reading  of  an  angle.  The  objection  ordinarily 
urged  against  such  a  procedure  is  that  it  fails  to  furnish  a  suffi- 
cient guard  against  mistakes  in  reading.  The  present  practice 
of  the  Coast  and  Geodetic  Survey  is  to  consider  that  a  pair  of 
readings  on  each  signal,  one  with  the  telescope  in  the  direct  po- 
sition, and  the  other  with  it  in  the  reverse  position,  together  con- 
stitute one  observation,  and  to  shift  the  position  of  the  limb 
before  the  next  observation.  A  comparison  of  the  direct  and 
reverse  readings  furnishes  a  rough  method  of  detecting  mistakes 
in  reading. 

65.  Assignment  of  Weight  Arbitrarily.  —  So  far  we  have 
deduced  the  combining  weights  from  the  observed  values  them- 
selves, or  from  them  in  connection  with  a  special  series  of  obser- 
vations. But  this  may  not  always  be  the  best  way  of  finding  the 
weights.  The  observations  may  not  be  our  only  source  of 
information,  and,  indeed,  not  the  most  reliable  source.  If,  for 
example,  some  phenomenon  has  been  observed  by  many  persons 
in  different  parts  of  the  country,  and  the  observations  are  sent 
to  one  place  for  comparison  and  reduction,  it  would  not  be  proper 
for  the  computer  to  deduce  a  weight  for  each  series  from  the 
observations  themselves  independent  of  other  sources  of  infor- 
mation he  might  have.  Some  of  the  most  inexperienced  obser- 
vers with  the  poorest  instruments  might  have  apparently  better 
results  than  the  most  experienced  with  good  instruments.  In 
such  a  case  the  computer  must  exercise  his  own  judgment  in 
classing  the  observations.      He  should  consider  the  experience  of 


OBSERVATIONS   OF    ONE    UNKNOWN    QUANTITY        83 

the  obsen'cr,  his  pre\nous  record  for  accurate  work,  the  kind 
of  instrument  used,  the  conditions,  and  the  observer's  record 
of  what  he  saw  —  whetlier  it  is  clear  and  precise  or  hazy  in  its 
statements.  An  arbitrary  scale  of  weights  may  then  be  con- 
structed, and  to  each  set  of  observations  be  assigned  a  weight 
from  this  scale  according  to  the  computer's  estimate  of  its  value. 
No  two  computers  would  be  likely  to  assign  precisely  the  same 
weights,  but  if  done  by  one  of  experience  and  good  judgment, 
the  result  obtained  from  weighting  in  this  way  will  undoubted !>• 
be  of  more  value  than  that  found  by  the  strict  application  of  the 
formulas  of  least  squares. 

The  point  is  simply  this.  The  class  of  observations  considered 
may  be  expected  to  contain  systematic  errors  which  cannot  be 
determined,  and  is  therefore  not  capable  of  being  treated  by  the 
method  of  least  squares.  As  we  have  no  direct  means  of  elimi- 
nating this  kind  of  error,  we  must  do  so  indirectly  as  best  we 
can,  and  that  is  what  the  system  of  weighting  mentioned  seeks 
to  accomplish. 

An  example  will  be  found  in  the  discussion  of  the  Telescopic 
Observations  of  the  Transit  of  Mercury,  May  5-6,  1878,  Wash- 
ington, 1879,  where,  of  109  observations  sent  in,  to  only  18  was 
the  highest  weight  assigned.  Professor  Eastman,  under  whose 
direction  they  were  reduced,  says  :  "  .  .  .  Several  instances  may 
be  found  where  small  weight  is  given  to'  observations  that  appar- 
ently agree  well  with  those  to  which  the  highest  weight  is 
assigned,  but  in  most  cases  the  observer's  remarks  indicate  the 
uncertain  character  of  the  observation." 

66.  Combination  of  Good  and  Inferior  Work.  —  It  is 
strictly  in  accordance  with  the  idea  of  weight  that  if  we  have 
two  results  of  very  different  degrees  of  accuracy,  a  result  better 
on  the  whole  than  either  may  be  found  by  combining  both  with 
their  proper  weights.  But  the  proper  weights  may  be  difficult 
to  find.  On  this  account  it  depends  on  circumstances  whether 
it  is  advisable  to  reduce  a  set  of  observations  poorly  made,  in 
order  to  combine  them  with  a  well-made  set.     If  the  quantity  is 


84  THE    ADJUSTMENT    OF    OBSERVATIONS 

available  for  observing  again,  it  might  not  cost  any  more  to  do 
this  than  to  reduce  the  poor  observations.  Even  if  it  did,  the 
residt  would  be  more  satisfactory.  The  committee  of  the  Royal 
Society  of  England  which  was  appointed  to  examine  Col.  Lamb- 
ton's  geodetic  work  in  India  reported  that  "  Col.  Lambton's  sur- 
veys, though  executed  with  the  greatest  care  and  ability,  were 
carried  on  under  serious  difficulties,  and  at  a  time  when  instru- 
mental appliances  were  far  less  complete  than  at  present. 
There  is  no  doubt  that  at  the  present  time  the  surveys  admit  of 
being  improved  in  every  part.  The  standards  of  length  are  bet- 
ter ascertained  than  formerly,  and  all  uncertainty  on  the  unit  of 
measure  may  be  removed.  The  base-measuring  apparatus  can 
be  improved.  The  instruments  for  horizontal  angles  used  by 
Col.  Lambton  were  inferior  to  those  now  in  use.  .  .  .  The  com- 
mittee express  the  strong  hope  that  the  whole  of  Col.  Lambton's 
survey  may  be  repeated  with  the  best  modern  appliances."  * 

67.  The  Weight  a  Function  of  our  Knowledge.  —  If  a 
quantity  is  not  available  for  observing  again,  as,  for  example, 
some  transient  phenomenon,  all  of  the  material  on  hand  must  be 
used,  and  the  best  weights  possible  assigned  to  the  separate 
values  in  order  to  combine  them.  The  point  is,  that  where  sys- 
tematic or  constant  error  has  not  been  eliminated,  the  weight  to 
be  assigned  is  a  function  of  the  state  of  our  knowledge  —  is,  in 
fact,  a  matter  of  individual  judgment. 

This  is  brought  out  very  fully  in  the  methods  used  in  com- 
bining the  older  star  catalogues  with  the  more  modern  ones. 
Thus,  Safford  (Catalogue  of  Mean  Declinations  of  20\Z  Stars, 
Washington,  1879)  says  :  "  In  computing  positions  I  have  gen- 
erally employed  Argelander's  rule,  giving  to  a  modern  determi- 
nation from 

1  observation  a  weight  ^, 

2  observations  a  weight  f, 

3  to  8  observations  a  weight  i, 

9  or  more  observations  a  weight  i^  or  2. 
*  G.  T.  Siirvey  of  India,  vol.  ii.  p.  70. 


OBSERVATIONS    OP    ONE    UNKNOWN    QUANTITY        85 

Argelander  generally  gives  Piazzi  a  weight  equal  to  unity ;  the 
value  ^  is  much  nearer  the  truth  ;  in  general  he  assigns  rather  a 
larger  relative  weight  to  the  older  and  poorer  observations  than 
they  deserve.  But  this  is  mostly  compensated  for  by  the  num- 
ber of  determinations." 

The  weight  of  a  quantity  being  a  function  of  our  knowledge 
may  have  assigned  to  it  a  certain  value  at  one  time  and  another 
value  at  another  time  when  our  knowledge  of  it  has  increased. 
Thus,  in  the  Fond  du  Lac  (Wis.)  base  of  the  Lake  Survey, 
measured  in  1872  with  the  Bache-Wurdemann  compensating 
apparatus,  a  portion  was  measured  seven  times.  The  results 
differed  widely,  far  beyond  what  was  expected  with  the  appar- 
atus. No  reason  could  be  assigned  at  the  time  for  the  discor- 
dances. At  this  stage,  then,  one  would  have  been  justified  in 
assigning  a  small  weight  to  the  value  of  the  base. 

The  Keweenaw  base  was  next  measured  with  the  same 
apparatus,  and  the  same  trouble  came  in.  Next  the  Sandy 
Creek  base  and  then  the  Buffalo  base  were  measured.  During 
all  this  time  (four  years)  material  had  been  accumulating  for  the 
explanation  of  the  behavior  of  the  apparatus.  When  the  law  of 
its  behavior  was  discovered,  it  was  found  that  good  work  not 
only  could  be  done  but  had  been  done  with  it. 

Hence  the  systematic  error  being  got  rid  of,  one  would  be 
justified  in  increasing  the  weights  of  the  bases  measured  with 
this  apparatus  in  comparison  with  bases  measured  with  an 
apparatus  of  a  different  kind.  Had  the  later  work  not  been  done, 
the  Fond  du  Lac  base  would  still  have  had  assigned  to  it  the 
low  weight. 

Take  another  instance.  Sir  G.  B.  Airy,  in  1847,  says  of  the 
Mason  and  Dixon  arc  {Encyc.  Mctrop.,  p.  209)  :  "The  results  of 
this  measure  must,  we  think,  be  received  as  equal  in  authority 
to  those  of  any  other  measure."  This  may  have  been  true 
when  written;  but  Mr.  Schott,  in  1877,  in  his  note  on  the 
determination  of  the  figure  of  the  earth  from  American  sources, 
says  of  this  same  arc  {U.  S.  C.  S.  Report,  1877,  p.  95):  "  It  is, 


86  THE    ADJUSTMENT    OF    OBSERVATIONS 

therefore,  only  owing  to  the  increased  perfection  of  instrumental 
means  and  methods  that  we  now  dismiss  from  further  considera- 
tion the  first  measured  North  American  arc,  which,  moreover, 
is  now  superseded  by  the  present  measures." 

As  a  third  illustration  we  may  consider  the  weights  to  be 
assigned  to  a  system  of  differences  of  longitudes  in  which  the 
connections  of  the  stations  occupied  are  interlaced  as  in  a  trian- 
gulation  net,  and  the  whole  system  is  to  be  adjusted  so  as  to 
remove  existing  contradictions. 

If  the  longitude  work  has  been  carried  out  on  one  plan,  with 
instruments  and  observers  of  about  the  same  quality,  then  the 
m.  s.  e.  of  each  determination  may  be  computed  from  the  mea- 
sures of  the  separate  nights,  and  in  the  adjustment  the  weights 
may  be  taken  inversely  as  the  squares  of  these  m.  s.  e. 

But  if  this  has  not  been  done,  if  in  the  older  work  instruments, 
observers,  and  methods  were  poorer  than  later  and  the  two  have 
to  be  combined  in  the  adjustment,  the  computer  must  estimate 
as  best  he  can  their  relative  weights.  Thus,  in  a  system  in  Ger- 
many, France,  and  Austria  reduced  by  Dr.  Albrecht*the  obser- 
vations were  made  between  the  years  1863  and  1876.  The 
methods  of  observation  had  been  much  improved  in  this  interval. 
In  assigning  the  relative  weights,  a  scale  of  weights  was  first 
formed  from  a  consideration  of  all  the  knowledge  on  hand,  tak- 
ing the  march  of  improvement  from  year  to  year  into  account, 
and  the  separate  determinations  placed  in  one  or  other  of  these 
classes.     Thus,  for  example, 

Weight  I,  —  No  change  of  observers;  few  observations;  non- 
adjustment  of  electric  current ; 
Weight  2,  —  No  change  of   observers ;  usual   variety  of  obser- 
vations ;  non-adjustment  of  electric  current ; 
Weight  3,  —  Change  of  observers  ;  usual  variety  of  observations  ; 

non-adjustment  of  electric  current, 
and  so  on. 

Similarly    Dr.    Bruhns  in     Vcrhandhingejt    der   europdischen 
*=  Astro7io>iiische  NachricJiten,  2132. 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         87 

Gradmessung,   1880.      See  also    Coast    Siin>ey    Report,    1880, 
Appendix  6  ;    1897,  Appendix  2. 

68.  General  Remarks.  —  The  subject  of  the  weightino-  of 
observations  is  confessedly  a  difficult  one.  In  general  it  may 
be  affirmed  that  the  less  experienced  a  computer  is,  the  more 
closely  he  will  adhere  to  the  rigorous  formulas  without  consider- 
ing whether  systematic  errors  enter  or  not.  As  he  adds  to  his 
experience  he  will  consider  outside  evidence  as  well  as  the  evi- 
dence afforded  by  the  observations  themselves.  This  will  be 
specially  true  if  he  has  any  practical  knowledge  of  how  observa- 
tions are  made.  Indeed,  it  is  doubtful  if  a  computer  can  apply 
the  principles  of  least  squares  properly  unless  he  is  at  least  an 
average  observer. 

Of  the  Rejection  of  Observations. 

69.  There  is  nothing  in  the  whole  theory  of  errors  more  per- 
plexing than  the  question  of  what  shall  be  done  with  an  obser- 
vation of  a  series  which  differs  widely  from  the  others.  In 
making  a  series  of  observations  the  observer  is  given  full  power. 
He  can  vary  the  arrangements,  choose  his  own  time  for  working  ; 
he  can  do  anything,  in  fact,  that  in  his  best  judgment  will  tend 
to  give  the  best  value  of  the  observed  quantity.  But  when  he 
has  finished  observing  and  goes  to  computing,  has  he  the  same 
power.!*  Can  he  alter,  reject,  manipulate  in  such  a  way  as  in  his 
best  judgment  will  give  a  result  of  maximum  probability.?  As 
observer  he  was  supreme  ;  as  computer  is  he  supreme,  or  only 
in  leading-strings  .?  Various  answers  may  be  given  to  this 
question,  as  we  look  at  it  from  one  point  of  view  or  another. 
When  observations  are  made  by  one  man  as  an  expert  ob- 
server, and  reduced  by  another  as  an  expert  computer,  the  judg- 
ment of  each  should  be  authoritative  in  his  own  province.  The 
observer's  statements  of  fact  as  to  the  conditions  under  which 
the  observations  were  made  must  be  accepted.  His  statements 
of  opinion  as  to  the  effects  of  such  conditions  on  the  accuracy 
of  the  obser\'ations  must  be  given  great  weight,  and  only  set 


88  THE    ADJUSTMENT    OF    OBSERVATIONS 

aside  when  a  considerable  mass  of  carefully  considered  evidence 
indicates  that  the  opinion  is  not  correct. 

On  the  other  hand,  in  judging  as  to  the  accuracy  of  a  particu- 
lar observation  or  group  of  observations  by  the  residuals  alone, 
the  judgment  of  the  computer  is  authoritative  rather  than  that 
of  the  observer.  He  has  in  general  a  much  wider  acquaintance 
than  the  observer  with  the  law  of  distribution  of  error,  as  he 
deals  during  his  regular  routine  with  the  observations  of  many 
different  men  made  under  many  conditions  and  on  different 
kinds  of  work.  The  observer  is  apt  to  form  his  opinion  of  a 
given  observation,  in  so  far  as  he  judges  it  by  the  corresponding 
residual,  by  comparing  it  with  the  preceding  observations  in  the 
same  series.  The  computer  bases  his  judgment  on  all  the  ob- 
servations, preceding  and  following. 

70.  There  is  one  respect  in  which  experience  shows  that  the 
opinion  of  the  observer  is  frequently  erroneous.  As  he  sees  the 
conditions  under  which  observations  are  made,  vary  from  those 
extremely  favorable  to  accurate  measurement  to  those  extremely 
unfavorable,  he  more  or  less  explicitly  assigns  to  the  observa- 
tions varying  weights.  The  range  of  weights  assigned  is  as  a 
rule  much  too  large.  If  he  believes  that  the  observations  made 
under  the  most  unfavorable  conditions  should  be  given  i  as 
much  weight  as  those  under  the  best  conditions,  the  chances  are 
that  the  ratio  of  the  true  weights  is  more  nearly  i  to  i.  In 
other  words,  the  observer's  best  observations  are  usually  poorer 
than  he  believes  them  to  be,  and  his  poorest  better.  He  is  mis- 
led by  his  feelings,  and  estimates  the  accuracy  of  the  obser- 
vations by  the  difficulty  of  securing  them  rather  than  by  a  care- 
ful systematic  study  of  them  in  the  light  of  all  available  evidence. 
The  observer  is  therefore  in  general  too  apt  to  reject  obser- 
vations made  under  difficult  conditions  or  to  decline  to  observe 
under  such  conditions. 

71.  In  the  hypothetical  case  on  which  the  exponential  law  of 
error  was  founded,  there  were  no  discontinuous  observations 
taken  into  account.     There  we  contemplated  not  only  observa- 


OBSERVATIONS    OF    ONE    UNKNOWN    QUANTITY         89 

tions  made  with  the  best  instruments  and  by  the  most  experi- 
enced observers,  but  observations  of  all  grades,  from  this  highest 
grade  down  to  those  made  with  the  poorest  instruments  and  by 
the  most  ignorant  and  careless  observers  conceivable.  It  is 
only  in  this  way  that  errors  continuous  all  the  way  from  -f  co  to 
—  00  could  arise.  In  the  cases  occurrmg  in  ordinary  work  we 
confine  our  attention  to  one  section  of  the  observations  only  — 
that  made  with  the  good  instrument  and  by  the  skillful  observer. 
This,  to  be  sure,  is  the  most  important,  and,  as  shown  in  Art. 
22,  the  result  following  from  it  differs  ordinarily  but  little  from 
that  found  in  the  ideal  case.  But  we  are  naturally  confronted 
with  difficulty  when  we  try  to  deal  with  a  very  incomplete  series. 
Extra  assumptions  must  be  made,  and  it  is  not  to  be  wondered 
at  that  no  solution  yet  offered  is  regarded  as  entirely  satis- 
factory. 

When  it  is  proposed  to  reject  a  certain  observation,  it  should 
be  kept  clearly  in  mind  that  the  only  justification  for  rejection 
is  that  by  so  doing,  (i)  the  effect  of  a  blunder  may  be  eliminated 
from  the  final  result,  (2)  or  the  effect  of  some  error,  from  an 
unusual  source,  of  much  greater  magnitude  than  the  errors 
affecting  the  other  observations,  may  be  eliminated.  The  essen- 
tial difficulty  in  deciding  what  to  reject  is  encountered  in  decid- 
ing whether  a  given  large  residual  is  due  to  either  of  the  causes 
indicated,  or  is  merely  due  to  the  accidental  agreement  in  sign 
of  many  small  errors  from  various  sources,  and  is  in  conformity 
with  the  law  of  error.  If  the  residuals  are  in  strict  accordance 
with  the  law  of  error,  there  will  be  a  few  which  stand  out  rather 
far  beyond  the  general  range.  (See  table  i.)  If  the  observations 
corresponding  to  these  are  rejected,  the  final  result  is  in  general 
reduced  in  accuracy  rather  than  increased.  Various  criterions 
for  rejection  have  been  devised  and  used,  some  of  them  being 
rather  complicated  as  to  their  theoretical  basis  and  tedious  in 
application.  The  following  criterion  commends  itself  as  Ining 
simple,  quick  of  application,  and  upon  a  sufficiently  good  theoret- 
ical basis.      It  is  recommended  for  general  use. 


90        THE  ADJUSTMENT  OF  OBSERVATIONS 

Reject  each  observation  for  zvJiich  the  residual  exceeds  five  times 
the  probable  error  of  a  single  observation.  Examine  each  observa- 
tion for  ivhich  the  residual  exceeds  3^  times  the  probable  error  of 
a  single  observation,  and  reject  it  if  any  of  the  conditions  under 
luhich  the  observation  ivas  made  zvere  such  as  to  produce  any  lack 
of  confidence. 

The  theoretical  basis  of  the  rule  is  evident  from  an  inspection 
of  table  I .  If  the  residuals  follow  the  law  of  error,  but  one  resid- 
ual in  55  should  exceed  3I  times  the  probable  error  of  a  single 
observation,  and  but  i  in  1000  should  exceed  5  times  that  value. 
The  presumption  is  strong,  therefore,  that  rejections  made  under 
the  rule  are  justified  in  most  cases.  A  few  observations  will  be 
rejected  by  this  rule  which  should  be  retained,  but  only  a  few, 
and  therefore  little  damage  will  be  done  in  any  case. 

The  probable  error  for  use  in  the  above  rule  should,  of  course, 
be  computed  from  all  the  observations  not  rejected  up  to  the 
time  of  the  proposed  application. 

72.  Rejected  observations  should  be  left  in  the  record  and 
computation  and  in  the  publication,  being  simply  marked  Re- 
jected. The  facts  then  appear  for  the  inspection  of  those  who 
follow,  and  may  serve  as  a  basis  for  indepeildent  conclusions. 

In  examining  a  large  residual,  it  will  sometimes  appear  that  it 
is  so  evidently  due  to  a  "natural  mistake"  that  it  may  be  cor- 
rected without  a  doubt  from  the  evidence  furnished  by  the  other 
observations,  and  the  discrepant  observation  changed  so  that  it 
may  be  treated  as  a  good  one.  Thus,  an  angle  may  be  read  5' 
or  10'  wrong,  or  a  micrometer  screw  may  be  read  5  or  10  revo- 
lutions out  of  the  way,  as  shown  by  the  rest  of  the  observations ; 
and  the  like.  Such  corrections  should  be  made  with  great 
caution,  however,  especially  if  the  number  of  observations  is 
small.  In  the  precise  leveling  of  the  Coast  and  Geodetic  Sur- 
vey, the  observer  is  required  to  run  over  each  section  of  the  line 
twice.  If  a  discrepancy  between  two  results  is  discovered  that 
is  greater  than  is  allowable  and  which  is  evidently  due  to  a  nat- 
ural mistake,  the  observer  is  not  allowed,  no  matter  how  plain 


OBSERVATIONS    OF    OXE    UNKNOWN    QUANTITY        91 

the  case  may  be,  to  correct  the  mistake  and  continue.  He  must 
rerun  the  section  until  he  secures  two  results  which  are  within 
the  required  limit  and  are  not  subject  to  any  assumption  as  to  a 
natural  mistake. 

73.  Again,  the  computer,  instead  of  trusting  to  his  judgment, 
may  call  in  the  aid  of  the  calculus  of  probabilities,  and  seek  to 
establish  a  test  or  criterion  for  the  rejection  of  observations 
which  will  serve  for  all  kinds  of  observations.  Of  the  criterions 
which  have  been  proposed  the  earliest  is  due  to  Professor  Peirce. 
It  is  as  follows:  "Observations  should  be  rejected  when  the 
probability  of  the  system  of  errors  obtained  by  retaining  them  is 
less  than  that  of  the  system  of  errors  obtained  by  their  rejection 
multiplied  by  the  probability  of  making  so  many  and  no  more 
abnormal  observations."  A  proof  by  Dr.  Gould  will  be 
found  in  the  U.  S.  Coast  Siwvey  Report,  1854,  pp.  131,  132. 
It  is  founded  on  the  assumption  of  the  Gaussian  law  of 
error. 

Another  criterion  "for  the  rejection  of  one  doubtful  observa- 
tion" is  given  by  Chauvenet  in  his  Astronomy,  vol.  ii.  p.  565. 
"We  have  seen  that  the  function  (Art.  30) 


2 


IT  Jo 


represents  in  general  the  number  of  errors  less  than  a  which 
may  be  expected  to  occur  in  any  extended  series  of  obser- 
vations when  the  whole  number  of  observations  is  taken  as 
unity,  r  being  the  p.  e.  of  an  observation.  If  this  be  multi- 
plied by  the  number  of  observations  ;/,  we  shall  have  the 
actual  number  of  errors  less  than  a  ;  and  hence  the  quantity 

n  -  )i®(l)  =  n{i  -  ©(/)} 

expresses  the  number  of  errors  to  be  expected  greater  than 
the  limit  a.  l^ut  if  this  C|uantity  is  less  than  ]  it  will  follow  that 
an  error  of  the  magnitude  a  will  have  a  greater  ]-)robability 
against   it    than  for  it,  and  may,  therefore,  be  rejected.     The 


92  THE    ADJUSTMENT    OF    OBSERVATIONS 

limit  of  rejection  of  a  single  doubtful  observation  is,  therefore, 
obtained  from  the  equation 

i  =  ;/{i  -©(/)} 

or  0  (/)  = 


A  third  criterion  was  proposed  by  Mr.  Stone,  Radcliffe  observer 
at  Oxford,  Eng.,  in  Month.  Not.  Roy.  Astron.  Soc,  i868,  1873, 
in  these  terms  :  "I  assume  that  a  particular  person,  with  definite 
instrumental  means  and  under  given  circumstances,  is  likely  to 
make,  on  an  average,  one  mistake  in  the  making  and  registering 
n  observations  of  a  given  class.  The  probability,  therefore,  is 
that  any  record  of  his  of  this  class  of  observations  as  a  mistake 

is  -.     From    the  average   discordances   among    the    registered 

observations  of  this  class  we  can  find  the  p.  e.  of  an  observation 
in  the  usual  way,  and  also  the  probability  of  an  error  greater 
than  a  given  quantity,  as  C.     Then  if  the  probability  in  favor  of 

a  discordance  as  large  as  C  is  less  than  that  of  a  mistake,  or  -> 

the  discordant  observation  is  rejected." 


CHAPTER    IV. 

ADJUSTMENT   OF    INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN. 

Determination  of  the  Most  Probable    J^a/ites. 

74.  If  direct  measurements  of  a  quantity  have  been  made 
under  the  same  circumstances,  we  have  seen  that  the  arithmetic 
mean  of  these  measures  gives  the  most  probable  value  of  the 
quantity.  We  now  come  to  the  case  where  the  quantity  meas- 
ured is  not  the  unknown  required,  but  is  a  linear  function  of  one 
or  more  unknowns  whose  values  are  to  be  found.  This  is  the 
more  general  form. 

Let  the  equations  connecting  a  series  of  observed  quantities 
J/,  J/„  .  .  .  J/„,  ;/  in  number,  and  the  independent  unknowns 
X,    F,  .   .  .  ,  n,  in  number  (n  >  n,),  be 

a,X  -\-b,Y  +  •  •  •  -  L,  =  Ml  +  v^, 

a^X  +  &,K  +  .  •  •  -  L2  =  M2  +  v^,  (i) 


a^X  +  6„F  +  •  •  •  -  L„  =  M„  +  v^, 
where  a^,  b^,  .   .  .  L^.  .  .  L^  are  constants  given  by  theory  for 
each  observation,  and  v^,  v^,  .  .  .  v„  are  the  residual  errors  of 
observation. 

In  practice  the  labor  of  handling  these  equations  will  be  much 
lessened  by  using  an  artifice  we  have  several  times  already  em- 
ployed (see  Art.  26).  Let  X',  Y',  .  .  .  be  close  approximations 
to  the  value  of  A',  K,  .  .  .  found  by  ordinary  elimination  from  a 
suflficient  number  of  the  equations,  or  by  some  other  method,  as 
by  trial,  for  example,  and  put 

X-r  =  x,Y  -Y'  =  y,-  ■  ' 
where  x,  y,  .  .  .  are    the    corrections    required    to  reduce   the 
approximate  values  to  the  most  probable  values. 

93 


94  'i^HE    ADJUSTMENT    OF    OBSERVATIONS 

Then  the  observation  equations  reduce  to 

a2X  -\-  b^y  -\-  •  '  •  —  l^^v^,  (2) 

QnX  +  b„y  +  '  ■  •  —  /„  =  t*,, , 
where 

-l,=  a,X'  +  b,Y'  +  .  ■  •-L.-M^, 
—  l2  =  a^X'  +  b^Y'  +  »  .  •  -  L3  -  M2, 


-  /„  =  a,X  +  bX  +  •  .  .  -  L„  -  M,„ 

and  are,  therefore,  known  quantities. 

Since  the  principle  that  the  sum  of  the  squares  of  the  residual 
errors  is  a  minimum  holds  whether  the  observed  quantity  is  a 
function  of  one  or  of  several  unknowns  (Art.  12),  we  can  apply 
it  to  the  simultaneous  solution  of  the  equations. 

The  residual  errors  must  satisfy  the  relation 

Vi'  +  v^  +  '  •  •  +  v,^-  =  a  min. ; 
that  is,  we  must  make 

(a^x  +b,y-^-  ■  •  -  /i)2  +  (a,.v  +  h.j  +  -  -  -  -l,f 

+  •  •  •  +  (a„x  +  b„y  -f-  .  .  .  —  /„)2  =  a  min. 

Now,  the  variables  x,  j/,  .  .  .  being  independent,  the  differential 
coefficients  of  the  expression  for  the  minimum  with  respect  to 
each  of  them  in  succession  must  be  equal  to  zero. 
Hence 


aiia^x  +  &,>'+•  •  •  -  li)  +  a2(^'^2^'  +  b.y  -\-  ■ 
bi(aiX  +  b^y  +  •  •  •  —  li)  +  b,(a^x  +  b^y  +  ■ 

•  •  -  /i)  +  •  •  •  =  0, 
••-/,)  +  ...  =  0,  (3) 

or,  collecting  the  coefficients  of  x,  y,  .  . 

[aa]  X  +  [abl  y  -f  [ac]  3  +  •  • 
[ab]x+  [bb']y  +  [bc]z  +  ■  ■ 
\ac\  X  +  [be]  y  +  [cc]  z  +  ■  ■ 

.  in  each  equation, 
•  =  [all 

■  =  m,           (4) 

■  =  [ell 

INDIRECT    OBSERVATIONS    OF    ONE     UNKNOWN         95 

These  equations  are  called  noj'mal  equations.  The  number 
of  these  equations  is  the  same  as  the  number  of  unknowns; 
that  is,  7ii.  Their  solution  will  give  the  most  probable  values 
of  X,  y,  .  .  .,  and  adding  these  values  to  the  approximate  values 
X',  Y'  .  .  .  already  known,  the  most  probable  values  of  X, 
F,  .  .  .  will  result. 
It  is  useful  to  2iotice  that  equations  (3)  may  be  written 

[<rc']  =  o, 

W  =  o,  (5) 

These  relations  correspond  to  [^']  =  o  in  the  case  of  the  arith- 
metic mean,  and  may  be  used  as  a  check  on  the  computation  of 
the  values  of  the  residuals. 

Ex.  —  Given  the  elevation  of  Ogden  above  the  ocean  by  C.  P.  R.  R.  levels 
to  be  4301  feet,  and  the  elevation  of  Cheyenne  to  be  6075  feet ;  also  the  ele- 
vation of  Cheyenne  above  Ogden  by  U.  P.  R.  R.  levels  to  be  1749  feet;  find 
the  adjusted  elevations  of  Ogden  and  Cheyenne  above  the  ocean,  supposing 
the  given  results  to  be  of  equal  value. 

Let  X,  V  denote  the  elevations  of  Ogden  and  Cheyenne  respectively. 
Then 

{X  -  4301)2  +  {y-  6075)'  +  (X  -V+  1749)'  =  a  min. 
Differentiate  with  respect  to  X,  Fin  succession,  and 

2X  -  V=  2552 
-  X+  2y=  7824. 
.•.  X  =  4309  feet. 
V  =  6067  feet. 

75.  If  the  observation  equations  are  of  different  weights 
/ ,  />.„  .  .  .  />,„  then,  reducing  each  equation  to  the  same  unit  of 
weight  by  multiplying  it  by  the  square  root  of  its  weight,  we 
have  (Art.  48), 

yfp^a^x  +  ^fpAy  +  •  ■  •  -  ^Ji  =  ^/7l^l» 

y/p'^a.x  +  ^JXy  +  •  •  •  -  ^Ji  =  ^/^'^'2'  (0 


with 


^PrflnX  +    ^nKy  +  ■    ■    •  -    ^Jn  =    "^iK> 

[fv-]  =  a  min. 


96  THE    ADJUSTMENT    OF    OBSERVATIONS 

Substituting  the  values  of  '^p^  v^,  'yp.ro.y,  ...  in  the  minimum 
equation,  and  differentiating  with  respect  to  x,  jj/,  ,  .  .  as  inde- 
pendent variables,  we  have  the  normal  equations 

[pan]  X  +  [pali]  y-{-  .  .  .  ^  [/,«/], 

{pah-\x  +  [pbb]y+.  ■  ■  =  ipbl-\,  (2) 


from  which  x,  y,   .   .  .  may  be  found. 

The  relations  for  weighted  equations  corresponding  to  those 
of  Eq.  3,  Art.  74,  are  evidently 

[pav]  =  o,  [pbi^]  =  o,.  .  .  (3) 

Formation  of  the  Normal  Equations. 

76.  Instead  of  forming  the  minimum  equation  and  differenti- 
ating with  respect  to  the  unknowns  in  succession,  it  is  more 
convenient  to  proceed  according  to  the  following  plans  suggested 
by  the  form  of  the  normal  equations  themselves. 

The  first,  hom.  equations  3,  Art.  74,  may  be  stated  as  follows  : 
To  form  the  normal  equation  in  x,  multiply  each  observation 
equation  by  the  coefficient  of  x  in  that  equation,  and  add  the  re- 
sults. To  form  the  normal  equation  in  y,  multiply  each  observa- 
tion equation  by  the  coefficient  oi  y  in  that  equation,  and  add  the 
results.     Similarly  for  the  remaining  unknowns. 

The  second  is  suggested  by  the  complete  form  of  the  normal 
equations  as  given  in  equations  4,  Art.  74.  According  to  this 
plan  we  compute  the  quantities  [^7^?],  [_c^b'],  .  .  .  [<?/],  etc.,  sep- 
arately, and  write  in  their  proper  places  in  the  equations. 

The  equality  of  the  coefficients  of  the  normal  equations  in  the 
horizontal  and  vertical  rows  leads  to  a  considerable  shortening  of 
the  numerical  work  in  computing  these  quantities.  Thus  with 
three  unknowns,  x,  y,  s,  all  the  unlike  coefficients  are  contained 
in 

+  [aa]  X  +  [ab]  y  +  [ac]  2  =  [«/], 

+  [bb']y+[bc]z  =  \bl], 

+  [cc]z=  [cl]. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN  97 

Instead,  therefore,  of  computing  12  quantities,  only  9  are  neces- 
sary, as  the  remaining  3  can  be  at  once  written  down.  With  n 
unknowns  the  saving  of  computation  amounts  to 

I  +  2  +  3  +  •  •  ■  +  {n-  i)  =  \n{n-  1) 
quantities. 

If  the  observation  equations  are  of  different  weights,  the  for- 
mation of  the  normal  equations  may  be  carried  out  precisely  in 
the  same  way  as  in  the  preceding  as  soon  as  the  observation 
equations  have  been  reduced  to  the  same  unit  of  weight. 

The  form  of  the  weighted  normal  equations,  however,  shows 
that  it  is  not  necessary,  in  order  to  obtain  the  coefficients  \^paa], 
\_pab^,  ...  to  multiply  the  observation  equations  by  the  square 
roots  of  their  weights,  and  form  the  auxiliary  equations  i.  Art. 
75,  since  the  products  aa,  ab,  .  .  .  multiplied  by  the  weights  of 
the  respective  equations  from  which  they  are  formed  and 
summed,  give  [pan'],  \_pab'],  .  .  .  directly.  This  is  important 
because  labor-saving. 

77.     Ex.  1.  —  Given  the  observation  equation.s,  all  of  equal  weight, 

X  =  I 

x+y  =3 

X  —  y  +  z  —  2 

-  X  —  y  +  z  =  I 

show  that  the  normal  equations  are 

4x  +  y  =5 

X  +  T,y  —  2  z  =  o 

-  2y  +  22  =  3 

Ex.  2.  — The  expansions  .r„  x.,,  x.,,  x,  for  1°  Fahr.  of  four  standards  of 
len.f^th  were  found  by  special  experiment  to  be  connected  by  the  following 
relations  at  a  temperature  of  62'  Fahr.  (m  =  one  micron.). 

^  X,  =       39-945     weight    i 


+    ^2                                =  5932 

^-  x^          =  5-371 

-I-     jfj  —  1.0937:1-3         =  0.006 

4-  4  OTj                      -  jr,  =  -    1 .335 

+  ;f ,                                 -  jr,  =  +  14.S33 


16 


find  their  most  probable  values. 


98 


THE    ADJUSTMENT    OF    OBSERVATIONS 


[The  normal  equations  are 


+  7  ^\                                         —  6x^  =  +  1 28. 943 

+  72.000,1-2—    8.750  A-3—  12^-4=  4-    78.940 

-    8.750  .n  +  13.569  .Ts  =+    21.432 

-  6.r,  -  12.000 .1-2                       +  gxi=  -    S4.993 


and  X,  =  39.913,  X.,  =  5.932,  x,  =  5.405,  x,  =  25.075.] 

An  example  will  now  be  given  to  illustrate  the  method  of 
forming  a  series  of  observation  equations  : 

Ex.  3.  —At  Washington  the  meridian  transits  of  the  following  stars  were 
observed  to  determine  the  correction  and  rate  of  sidereal  clock  Kessels  No. 
1324,  April  12,  1870,  at  11  hours  clock  time. 


Star. 

Observed  Clock 
Time  of  Transit,  T. 

Right  Ascension 
OF  Star, a. 

T  Leonis. 
V  Leonis. 
/3  Leonis. 
0  Virginis. 
7)  Virginis. 
d  Virginis. 

/t.         J>t.              S. 

II      21       17.98 
II      30      20.41 

11  42      2S.57 

u     58     38.15 

12  13        18.37 

13  3       16.36 

5. 
16.00 
18.51 

26.57 
36.20 
16.37 

14-39 

Let  x  =  corr.  of  clock  at  11  hours  clock  time, 

y  =  rate  per  hour  of  clock. 
Now,  from  theoretical  considerations  *  it  is  known  that  the  equation 

x+y  (T-ii)  =a~  T 

gives  the  relation  between  the  clock  correction  and  rate  and  the  clock  time 
of  transit  of  each  star  observed. 

Hence  the  observation  equations  are 


X  ^ 

0-35J' 

=  - 

1.98 

x  + 

0.50J 

=  - 

1.90 

x  + 

0.7 1  _y 

=  — 

2.00 

x  + 

o.98_>' 

=  - 

1.95 

x  + 

i.22y 

=  - 

2.00 

X  + 

2.osy 

=  - 

1.97 

For  the  remainder  of  the  solution,  see  Art.  114. 

Ex.  4.  —  In  the  triangulation  of  Lake   Superior  executed  by  the  U.  S. 
Engineers  there  were  measured  at  station  Sawteeth  East  the  angles 
*  See  Chauvenet,  Astronouiy^  vol.  ii.  chap.  v. 


INDIRECT    OBSERVATIONS    OP    ONE    UNKNOWN 


99 


62-"^  59'  40.33" 
64°  11' 34  92" 

weight  5 
"      7 

100°  20'  29.12" 

"      4 

37°  20'  49.55" 
36-  08'  55.86" 

"  7 
"      4 

Farquhar- Porcupine 

Farquhar-Outer 

Farquhar-Bayfield 

Porcupine-Bayfield 

Outer-Bayfield 


required  the  adjusted  values  of  the  angles. 

All  of  the  angles  may  evidently  be  expressed  in 
terms  of  FSP,  FSO,  FSB.  Let  X,  K  Z  denote 
the  most  probable  values  of  these  angles,  and  let 
X',  Y',  Z'  be  assumed  approximate  values  of 
these  most  probable  values,  and  x,y,  z  their  most 
probable  corrections.  Denoting  the  measured 
angles  in  order  by  iJ/,,  M^,  .  .  .  J/5,  and  their  most 
probable  corrections   by   7^,,  t'^,  .  .  .  7^5,  we  have 


X'  ^-  X  = 

Z'  +  z  = 
X'  -  X  +Z'  +  z  = 
Y'  -y  +  Z'  +  z  = 


X 
Y 
Z 

X  +  z 


=  J/,  +  z/, 
=  J/3  +  7/3 
=  M,  +  V, 


F!g.  3- 


-  K+Z  =  J/+  V, 


For  simplicity  the  assumed  approximate  values  may  be  taken  equal  to  the 
observed  values  of  the  angles,  so  that  we  have  the  reduced  observation 
equations 

+  X  =  7'i     weight  5 

+  y  =  V.2  "7 

+  z  =  V,  "       4 

—  X  +  z  —  0.76  =  7'4  "       7    - 

—  J/  +  £■  —  1.66  =  7/5  "       4 

Hence  the  normal  equations 

12. V  -    7s-=_    5.32 

+  1 1  J'  -    4  ^  =  —    6.64 

—  y  X  —    4J'  +  15 -^  =  +  11.96 

Solving  these  equations,  we  find 

X  =  —  0.05",      y  =  —  0.36",      z  =-V  0.68". 
Hence,  7/,  =  —  o  05",  7/.,  =  —  0.36",  7/.,  =  +  0.68",  7/4  =  —  0.03",  v,,  =  —  0.62", 

and  the  adjusted  values  of  the  angles  are 

62"^  59'  40.28" 
64°  n'  34.56" 
100°  20'  29.80" 
37^  20'  49.52" 
36°  08'  55.24" 


lOO  THE    ADJUSTMENT    OF    OBSERVATIONS 

We  might  have  used  7\,  v^,  .  .  .  7/5,  for  the  corrections  without  introducing 
the  symbols  .T,  J/,  s  at  all. 

Ex.  5.   If  the  unknown  x  occurs  in  each  of  tlie  ;/  observation  ec^uations 

-  X  +  b,y  ■\-  c,z  -V  .  .  .  =  I,  weight  1, 

—  X  +  lK_y  +  c.,s  +  .  .  .  =  4        "        I, 


these  equations  are  equivalent  to  the  reduced  observation  equations, 

-^iJ  +  ^1"  +  •  •  •  =  A     weight  I, 
b^_y  +  62^  +  .  .  .  =  /,  "       I, 


[^]j'+u->  +  .  ■  ■=in    "  -)• 


[For  the  normal  equations  found  from  the  first  set  after  eliminating  x  are 
the  same  as  the  normal  equations  formed  from  the  second  set  directly.] 

Ex.  6.  —  Instead  of  the  observation  equation 

ax  +  if  -\-  cz  +  .  .  .  =  /  weight  p, 
we  may  write 

gax  +  qby  +  .  .  .  =  ql  weight    ;,• 

78.    Control  of  the  Formation  of  the  Normal  Equations.  — 

A  very  convenient  check  or  control  is  the  following.  Addas  an 
extra  term  to  each  observation  equation  the  sum  of  the  coeffi- 
cients of  X,  y,  .  .  .  and  of  the  absolute  term  /  in  that  equation, 
and  treat  these  added  terms  just  as  we  do  the  absolute  terms. 
Thus  let  s^,  s,„  .   .  .  s,^^  denote  the  sums,  so  that 

ai  +  lh  +  Ci  +  ■  ■  •  +  /i  =  -^i ) 
a.,  +  b..  +  r.,  +  ••■+/.,  =  .V, , 


(Tn    +   ^'n  +  <'n   +    •••+/„=   S„ 


Multiply  each  of  these  expressions  by  its  a  and  add  the  prod- 
ucts, each  by  its  d  and  add,  and  so  on  ;  then 

[an]  +  [ah]  +  •  •  •  +  [al]  =  [as]  , 
[ab]  +  [bh]  +  ■  ■  -  +  [W]  =  [bs]  , 


[<//]  +  [/>/]  +  •  •  ■+[//]  =  [/.v]. 
If  these  equations  are  satisfied,  the  normal  equations  are  correct. 
Thus  each  normal  equation  is  tested  as  soon  as  it  is  formed. 

PROPERTY  OF 
MELVIN  D.  CASLFR, 

vnm  PT.ATN     N.  Y. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN       loi 

Since  [^?<7],  [rt<^]>  •  •  •  {_'^^^  have  been  computed  in  forming 
the  normal  equations,  the  only  new  terms  to  be  computed  in 
applying  the  check  are  \_as^,  [/^i-],   .  .  .    [/y],  [//J. 

Various  modifications  may  readily  be  applied  to  suit  individual 
tastes.  Thus  the  absolute  term  may  be  placed  on  the  other  side 
of  the  sign  of  equality ;  or  the  sign  of  the  check  may  be  changed 
so  as  to  make  the  sum  of  each  horizontal  row  equal  to  zero. 

79-  Forms  of  Computing  the  Normal  Equations.  —  When 
the  number  of  unknowns  in  the  observation  equations  is  large, 
or  when  their  coefficients  contain  several  figures,  it  is  convenient 
to  have  a  fixed  form  for  the  computation  of  the  terms  of  the  nor- 
mal equations.  It  lightens  the  labor  much  either  in  forming, 
solving,  or  in  finding  the  precision  of  the  unknowns  from  these 
equations,  if  the  computation  is  so  arranged  that  a  check  can  at 
all  times  be  applied  and  the  whole  process  proceed  in  a  uniform 
and  mechanical  manner. 

The  aids  in  the  arithmetical  work  are  a  table  of  squares,  a 
table  of  products  [Crelle's],  a  table  of  reciprocals,  a  table  of  log- 
arithms, and  an  arithmometer,  or  machine  for  performing  multi- 
plications and  divisions.  The  latter  is  of  the  greatest  use  in 
computations  of  this  kind.  With  it  the  drudgery  of  computation 
is  in  great  measure  got  rid  of.  On  the  Lake  Survey  two  forms 
of  machine  were  used,  the  Grant  and  the  Thomas.  In  the  Coa.st 
and  Geodetic  Survey  there  are  in  use  (1904)  the  Thomas  or 
Burckhardt  machine,  the  Brunsviga  machine,  and  the  Thatcher 
slide-rule. 

With  the  Crelle  multiplication  tables  as  good  speed  can  be 
made  as  with  the  machine  if  the  numl^cr  of  significant  figures 
required  in  the  products  is  so  small  that  little  or  no  interpolation 
is  required  in  the  tables. 

Form  {a).  With  Crelle's  tables,  or  with  a  machine,  the  prod- 
ucts an,  ah,  ...  are  found  directly,  and  all  that  is  then  to  be 
done  is  to  write  them  in  columns  and  take  their  sums  \jra\ 
\ab\  .  .  .  With  a  Thomas  machine,  however,  each  product  may 
be  added  to  all  that  precede,  so  that  the  final  values  result  at  once. 


102  THE    ADJUSTMENT    OF    OBSERVATIONS 

Let  us,  for  example,  take  the  observation  equations 

—  1.2  X  +  0.2  y  +  0.9  =  v^ , 
+  3.0.%-—  2.1  y+  i.i  =V2, 
-\-  o.'j  X  -{-  1.6  y  —  4.0  =  Vs  • 

Arrange  as    follows,  the    headings  indicating  the  nature  of  the 
numbers  underneath : 


a 

/; 

/ 

s 

—  1.2 

+  0.2 

+  0.9 

—  0.1 

+  3-0 

—  2.1 

+  I.I 

+  2.0 

+  0.7 

+  1.6 

-  4.0 

-  1-7 

aa 
1.44 
9.00 
0.49 

+  IO-93 

ab 

-  0.24 

-  6.30 
+  1. 12 
-5.42 

al 

-  1.08 

+  3-30 

-  2.80 

-  0.58 

as 
+  0.12 
+  6.00 
-  1.19 

+  4.93 

-  5-42 

bb 
0.04 
4.41 
2.56 

b/ 
+  0.18 
-2.31 
—  6.40 

-8.53 

bs 

—  0.02 

—  4.20 

—  2.72 
-6.94 

+  7.01 

-  0.58 

-8.53 

// 

0.81 
1. 21 

16.00 

/s 
—  0.09 
+  2.20 

+  6.80 
+  8.91 

+  18.02 

Hence  the  normal  equations,  with  the  check  all  ready  for  solu- 
tion, are 

0  =  4- 10.93.%'— 5. 42  J— 0.58          +4-93> 
o=—    5.42  a- +  7.01  J— 8.53          —6.94. 

80.    Fo7'm  {b).     If  logarithms  alone  are  used,  form  a  table  of 
the  log  coefficients  of  the  observation  equations  as  follows: 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN       103 

logaj,  log&i,  .  .  •  log/j,  logij, 
log  02,  log  6,,  •  •  •  log  /,,  log  5,, 


l0g(7„,  log/),,,  ■  •  .  log/,,,  log.v„. 


Write  log  a^  on  a  slip  of  paper  and  carry  it  along  the  top  row, 
forming  the  products, 

log  aifli,  log  ap^,  ■  ■  •  log  (/j/j,  log  a^s^ . 
Similarly  with  log  a.^  form  the  products  from  the  second  row, 
log  02^2'  ^og  «2^2'  •  •  •  log  a.J.,,  log  a,^' 

and  so  on  till  log  <t„  is  reached. 

The    numbers    corresponding   to    these  logarithms  are  next 
found,  so  that  we  have 

a^a^,  Oi&i,  •  •  •  a/i,  a^s^, 
a^a^.,  a.Jb.,,  ■  ■  ■  aj^,  a^s^, 


By  addition  we  find 

[aa],[ab],.  ■  •[<//],[./.], 

the  coefficients  of  the  unknowns  in  the  first  normal  equation. 

Proceed  in  a  precisely  similar  way  with  log  />^,  log  d.„  .  .  . 
ijj,  omitting  the  term  [^i^]  already  found  ;  with  log  r^,  log  c.„ 
.  .  .  log  c„,  omitting  the  terms  [<^r],  [^r]  already  found  ;  and  so 
on  till  the  last  quantity  is  reached. 

81.  Form  (c).  If  we  wish  to  use  a  table  of  squares  alto- 
gether, then,  since 

ab    =  ^{(a  +  by  -  a- -  b-} 
and  therefore 

M]  =  H[(«  +  /0'-']-M-W}  (0 

we  form  the  square  sums 


lo4 


THE    ADJUSTMENT    OF    OBSERVATIONS 


[aa],  [ia-{-by],[ia-\-cy], 

[bb],[(b  +  cy],. 


uiiw  +  m, 

and  perform  the  necessary  subtractions. 

In  doing  this,  first  take  from  the  table  of  squares  the  squares 
aa,  bb,  .  .  .  II,  ss,  and  sum  them  ;  next  write  the  coefficients  a 
of  X  on  a  sHp  of  paper  and  carry  them  over  the  coefficients  of  j, 
s,  .  .  .  ,  forming  the  sums  a^-^  b^,  a^  -[-  c^,  .  .  .  ;  a.,  +  b.„  a.,  4- 
c^,  .  .  .  Take  out  the  squares  of  these  numbers  and  sum  them. 
Proceed  similarly  with  the  coefficients  oi  j/,  s,  .  .  .  Finish  as 
indicated  in  (i). 

Thus  in  the  preceding  example, 


aa 

1.44 

9.00 

0.49 

10.93 

bb 

0.04 
4.41 
2.56 
7.01 

11 

0.81 
1. 21 

16.00 
18.02 

ss 

0.0 1 
4.00 
2.89 
6.90 

a  A-  b        (a 

+  bf 

a  +  /         {,a  +  If 

a  +  s 

{a  +  sf 

I.O 

1. 00 

0.3                 0.09 

1-3 

1.69 

0.9 

C.81 

4.1                16.81 

5.0 

25.00 

-•3 

5.29 

3.3                10.S9 

1.0 

1. 00 

7.10 

27.79 

27.69 

[aa]  +  Ibb]  = 

17.94 

[aa]  +  [//]  =  28.95 

[aa]  +  [ss] 

=  17-83 

— 

10.84 

-  1. 16 

9.86 

— 

5-42 

-  0.58 

4-93 

= 

[ab\ 

=  [al] 

=  [as] 

giving  the  same  results  as  before. 

This  form,  which  is  very  neat  analytically,  was  first  given  by 
Bessel  in  the  Astron.  Nachr.,  No.  399. 

A  consideration  of  the  simple  case  of  three  observation  equa- 
tions, each  involving  two  unknowns,  will  show  that  to  form  the 
normal  equations,  using  a  log  table  only,  24  entries  in  the  table 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN       105 

are  required,  while  by  this  method  we  only  need  to  enter  a  table 
of  squares  18  times,  thus  effecting  a  saving  of  6  entries.  The 
Bessel  method  has  also  the  advantage  that,  as  we  deal  with 
squares,  all  thought  with  regard  to  sign  is  done  away  with,  lie- 
sides,  if  the  table  of  squares  is  a  very  extended  one,  accuracy 
can  be  had  to  a  greater  number  of  decimal  places  than  with  an 
ordinary  log.  table.  As  compared  with  the  logarithmic  form, 
then,  this  method  is  to  be  preferred,  more  especially  when  the 
coefficients  are  not  very  different. 

On  the  other  hand,  if  Crelle's  tables  or  a  computing  machine 
is  to  be  had,  the  direct  process  explained  in  (a)  is  much  to  be 
preferred  to  either,  as  experience  will  show. 

82.  It  is  worth  noticing  that  whichever  method  of  formation 
of  the  normal  equations  is  adopted,  labor  will  be  saved  by  chang- 
ing the  units  in  which  the  unknowns  are  .expressed  if  the  coeffi- 
cients of  the  different  unknowns  are  very  different.  Thus, 
suppose  we  had  the  observation  equations. 

Check  Si"MS. 

1000  A"  +  o. 0001  y  =  4. 1 1  1004. 1  lOI 

999  .V  -|-  0.0C02  y  =  3-93  1002.9302 


from  which  to  find  x  and  j-. 
By  placing 

.v'  =100  X,  y   =  0.0 1  y 

the  equations  reduce  to 

Check  Sums, 
io.t'        -f- o.oi  y' =  4.1 1  14.12 

9.99  .v'  +  0.02  y'  =  3.93  13.94 


which  are  in  more  manageable  shape  for  solution. 

83.    Before  beginning  the  solution  of  a  scries  of  normal  ccjua- 
tions  we  should  consider  whether  the  object  is  to  find: 

(i)  the  unknowns  only,  or 

(2)  the  unknowns  and  their  weights; 


io6  THE    ADJUSTMENT    OF    OBSERVATIONS 

and,  in  the  latter  case, 

(a)  whether  the  number  of  unknowns  is  large, 

(b)  whether  many  of  the  coefficients  of  the  unknowns  in  the 
normal  equations  are  wanting. 

Normal  equations  may  be  solved  by  the  ordinary  algebraic 
methods  for  the  elimination  of  linear  equations  or  by  the  method 
of  determinants.  When,  however,  they  are  numerous,  the 
method  of  substitution  introduced  by  Gauss  and  the  Doolittle 
method  are  more  suitable.  Each  has  its  advantages.  Both  are 
quite  mechanical  in  operation  and  are  well  suited  for  use  with  an 
arithmometer,  which  is  as  great  a  help  in  solving  as  it  is  in 
forming  the  normal  equations. 

84.  The  Method  of  Substitution.  —  For  convenience  in  writ- 
ing, take  three  unknowns,  x,  j>,  z,  the  process  being  the  same 
whatever  the  number. 

The  normal  equations  are 

\ad\  X  +  [aft]  y  +  \ac\  z  =  [a/] , 

[aft].v+  [hb']y+  [6r]3  =  [W],  (i) 

[ac-]x+[bc-\y+[cc]z  =  [r/]. 

From  the  first  equation 

[aft]  \ac]      ,    [a/]  .  . 

[aaj  \_aa\  \_aa\ 

Substitute  this  value   in  the    remaining  equations,  and,  in  the 
convenient  notation  of  Gauss,  there  result 


where 


[ftft.i],v-f  [ftr.i]s=[ft/.i], 

[ftr.i]>'+[rr.i]z  =  [r/.i],  (3) 

[ftft.i]  =  [ftft]-[^][ai], 
[ft..i]=[ft.]-M[a.]. 
[W.I]  =  [ft/] -[^  [a/],  (4) 


(5) 


(6) 


(7) 


INDIRECT    OBSERVATIONS    OF   ONE    UNKNOWN       107 

[r/..]=[.v]-[^;M. 

Again,  from  the  first  of  equations  3, 

^^    [bb.iy '^  [bb.i]' 

which  value  substituted  in  the  second  equation  gives 

[r/.2] 

[CC.2] 

where 

Having  thus  found  :y,  we  have  j'  at  once  by  substituting  in   (5), 
and  thence  x  by  substituting  forj/  and  s  their  values  in  (2). 

The  first  equations  of  the  successive  groups  in  the  elimination 
collected  are 

[aa]  X  +  [ab]  y  -h  [ac]  z  =  [al]  , 

[W.i]v+[/^r.i]s  =  [W.i],  (8) 

{_rc.2]z  =  [cl.2]. 

These  are  called  the  derived  normal  equations. 

Divide  each  of  these  equations  by  the  coefficient  of  its  first 
unknown,  and 

M'     M"     M' 

y^\bbjr-\_bb.i]'  ^9^ 

'-[cC.2] 

85.  Controls  of  the  Solution.  —  In  solving  a  set  of  normal 
equations  a  control  is  essential.  It  is  sometimes  recommended 
to  solve  the  equations  arranged  in  the  reverse  order,  when,  if  the 


io8  THE    ADJUSTMENT    OF    OBSERVATIONS 

work  is  correct,  the  same  results  will  be  found  as  before.  But 
what  is  wanted  in  a  control  is  a  means  of  checking  the  work  at 
each  step,  and  not  at  the  conclusion,  it  may  be,  of  several  weeks' 
work,  when,  if  the  results  do  not  agree,  all  that  is  known  is  that 
there  is  a  mistake  somewhere  without  being  able  to  locate  it. 

(a)  Continuation  of  the  formation  control.  Experience  has 
shown  that  it  is  convenient  to  carry  on  through  the  solution  the 
check  used  in  forming  the  equations.  It  merely  consists  in 
placing  as  an  extra  term  to  each  equation  the  sums  [c?.s-],  {^ds'\, 
.  .  .  [A-],  and  operating  on  them  in  the  same  way  as  on  the 
absolute  terms  [«/],  [^^/]»  •  •  •  The  sum  of  the  terms  in  every 
line,  after  each  elimination  of  an  unknown,  must  be  each  equal 
to  the  check  sum  numerically  ;  the  closeness  of  the  agreement 
depending  on  the  number  of  decimal  places  employed. 

This  check  may  be  applied  at  every  step  and  mistakes  be 
weeded  out. 

(b)  The  diagonal  coefficients  [<^<^],  ['^'^J,  ...  of  the  normal 
equations,  and  [^aa^,  [<^/ai],  [rr.2],  ...  of  the  derived  normal 
equations,  are  always  positive. 

For  [f^?^?],  [/^^.  i],  .  .  .  being  the  sums  of  squares,  are  posi- 
tive.    Also 


r     n   TAT,    n        I  [c<^] '  [^^] 


a,,  ft. 


+ 


+ 


a  positive  quantity. 

(c)  By  equations  5,  Art.  74,  the  residuals  found  by  substitut- 
ing for  X,  y,  z  their  values  in  the  observation  equations  must 
satisfy  the  relations 

[,rc.]  =  [/r.]  =  .   .  .  =  o. 

(d)  A  very  complete  check  is  afforded  by  the  different 
methods  of  computing  \_vv'\  the  sum  of  the  squares  of  the  resi- 
duals.     (See  Art.  106.) 

86.  Forms  of  Solution.  — In  applying  the  method  of  substi- 
tution to  any  special  example  it  is  important  that  the  arrange- 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        109 

ment  of  the  computation  be  convenient  and  that  every  step  be 
written  down.  Experience  teaches  that  simpUcity  and  uniform- 
ity of  operation  are  great  safeguards  against  mistakes. 

Form  (a).      Solution  without  logarithms. 

The  following  form  has  been  found  by  experience  to  be  con- 
venient. It  is  well  fitted  for  use  with  the  arithmometer  or  any 
other  rapid  method  of  multiplication.  The  form  can  be  readily 
modified  to  suit  computer's  tastes. 

For  illustration  let  us  take,  as  before,  three  unknowns,  x,  y,  c. 
The  computation  is  divided  into  sections,  each  section  being 
formed  in  a  precisely  similar  way,  and  in  each  section  one  un- 
known is  eliminated. 

Given  the  normal  equations, 


No. 

X 

y 

z 

Check. 

Remarks. 

I. 
11. 
III. 

[aa] 
[a6] 
Wc] 

W6] 
\66] 
[6c] 

[ac] 
[6c] 
[cc\ 

[al] 
[61] 
[cl] 

[as] 
[6s] 
[cs] 

•     •     •     • 

Solution. 

IV. 

v. 

11. 
VL 

VH. 

III. 

VIII. 

IX. 

X. 

VIII. 

I 

[a6] 
[aa] 

[a6]   , 

[aa] 

[66] 
[66.1] 

[6c] 
[6c. i] 

[ac] 

[aa] 

[6c] 
[6c. ^] 

[cc] 

[al] 
[aa] 

r   AT  f"^ 

^'''^[a^\ 

[61] 

[61.x 
[cl] 

[f/.l] 

[as] 
[aa] 

r  Al  f"^' 

[6s] 

[6s.i] 

[cs] 
[cs.i] 

I. -=-[""] 

IV.  X  [a6] 
11. 

II. -v. 

IV.  X  [ac] 
III. 

III.- VII. 

V\.^[6fi.,] 
IX.X[^f.i] 

VIII.-  X. 

\l.-r\cc.2] 

[ccA 

I 

[6c. A 
[66.1] 

[cc.i] 

\6l.i\ 
[M.i] 

[cl.A 

[6s.,] 
[66.,] 

[cs.i] 

XI. 

[cc.i\ 

[CS.2] 

1 

[ci-A 

[.-r.2J 

{CS.2] 
]CC.2\ 

no  THE    ADJUSTMENT    OF    OBSERVATIONS 

...  2=  IeL^ 

From  Eq,  IX.  ^  _    [bc.i]       [bl.i] 

^~       ^[bb.i]^  [bb.i]' 

From  Eq.  IV,  ^^_     W_,M^W, 

[(w]  [aa]       [aa] 

To  eliminate  the  first  unknown,  x.     Tn  the  first  line  write  the 

quotients  J= — i,  -^ — i ,  .  .  .  that  is,  the  coefficients  of  the  first 

normal  equation  divided  by  [rf^r],  the  coefficient  of  x  in  that 
equation. 

The  first  line  is  now  multiplied  in  order  by  \_al?'],  \_ac'],  form- 
ing the  second  and  fifth  lines. 

In  the  third  and  sixth  lines  write  equations  II.  and  III. 

The  fourth  line  is  the  sum  of  the  second  and  third,  and  the 
seventh  the  sum  of  the  fifth  and  sixth. 

This  concludes  the  elimination  of  x,  and  the  results  in  the 
fourth  and  seventh  lines  involve  j  and  -c  only. 

Take  now  these  results  and  proceed  in  a  precisely  similar  way 
to  eliminate  J/. 

The  value  of  the  last  unknown,  ^,  next  results. 

Now  proceed  to  find  j/ and  x.  Thus,  substitute  for  s  its  value 
in  the  eighth  line,  and  we  have  j;  and  for  j  and  ^  their  values 
in  the  first  line,  and  we  have  x. 

87.  In  carrying  this  solution  into  practice,  there  are  three 
points  that  deserve  special  notice : 

(i)  In  order  to  render  the  work  mechanical,  and  so  lighten 
the  labor,  the  number  of  different  operations  should  be  made  as 
small  as  possible.  Instead,  therefore,  of  dividing  by  [aa], 
[d^.i],  [cc.2'],  it  is  better  to  multiply  by  the  reciprocals  of  these 
quantities,  and,  in  order  to  avoid  subtractions,  to  first  change  the 
signs  of  the  reciprocals.  We  shall  then  have  to  perform  only 
two  simple  operations  — multiplication  and  addition.  By  trans- 
ferring the  terms  [al],  [^/],  [^/]  to  the  left-hand  side  of  the 
equations  before  beginning  the  solution,  the  values  of  the  un- 
knowns will  come  out  with  their  proper  signs. 


INDIRECT    OBSERVATIONS  OF    ONE   UNKNOWN        iii 

(2)  Equations  VI.  and  \"III.  are  the  normal  equations  with  ;ir 
eliminated.  An  inspection  of  them  shows  that  the  coefficients 
of  the  unknowns  follow  the  same  law  as  the  coefficients  of  the 
unknowns  in  the  original  normal  equations  with  respect  to  sym- 
metry of  vertical  and  horizontal  columns.  Hence  in  the  elimi- 
nation it  is  unnecessary  to  compute  these  common  terms  more 
than  once.  Thus  \_l?c.i']  from  Eq.  VL  may  be  written  down  as 
the  first  term  of  Eq.  VIII.  This  principle  is  of  great  use  in 
shortening  the  work  when  the  number  of  unknowns  is  large. 

('3)  In  a  numerical  example  it  is  evident  that  since  ^aa'], 
\_dd.i'],  [cc.2]  do  not  in  general  divide  exactly  into  the  other  co- 
efficients of  their  respective  equations,  and  that  only  approxi- 
mate values  of  the  unknowns  can  at  best  be  obtained,  it  will  give 
a  closer  result  to  divide  by  the  larger  coefficients  and  multiply 
by  the  smaller  than  vice  versa.  Attention  to  this  by  a  proper 
arrangement  of  the  coefficients  before  beginning  the  solution  re- 
sults in  a  considerable  saving  of  labor,  as  the  successive  coeffi- 
cients in  the  course  of  the  elimination  need  not  be  carried  to  as 
many  places  of  decimals  to  insure  the  same  accuracy  that  a  dif- 
ferent arrangement  would  require. 

Ex. — To  make  the  preceding  perfectly  plain  we  shall  solve  in  full  the 
normal  equations  formed  in  Art.  79. 

(i)  Write  the  absolute  term  on  the  right  of  the  sign  of  equality,  and 
make  the  check  sum  equal  to  the  sum  of  the  other  terms  in  each  horizontal 


X 

y 

/ 

Check. 

Remarks. 

I. 
II. 

III. 

IV. 

II. 
v. 

VI. 
VII. 

III. 

VIM. 

+  >o93 
—    5-42 

+     ' 

-V    I 

-S.42 
-f-7.01 

—  0.496 

H-  2.688 
-j-7.01 

-1- 4-322 

—  0.496 

—  0.496 

4-0.58 
+  8.53 

+  0-053 

—  0.288 
+  8.53 

-f  8.818 

-|-  2.040=;;' 

—  1.012 
-1-  0.053 

f-  I  .C65  =  X 

+    6.09 
-f-  10.12 

+    0-557 

—  3-o'9 
■\-  10.120 

-V  13'39 
-t-    3-040 

—  1.508 
+    0.557 

-J-    2.0^.5 

I.    -r  10.93 

III.     X  -5-42 
11. 

II.    -  IV. 

V.    -r  4-322 

VI.    X— 0496 
III. 

Ill           \II. 

Hence 


X  =  1.06 


J  =2.04 


112 


THE    ADJUSTMENT    OF    OBSERVATIONS 


(2)  Write  the  constant  term  on  the  left  of  the  sign  of  equality,  and  form 
the  check  so  as  to  make  the  sum  of  the  terms  in  each  horizontal  line  equal 
to  zero. 


Reciprocals. 

- 

y 

/ 

Check. 

Remarks. 

I. 
II. 

0.09.5 

+  10.93 
—    5-42 

—  s-42 
+  7.01 

-0.58 
-8.53 

—  4-93 

+  6.94 

III. 

-    I 

+  0.496 

+  0.053 

+  0.451 

I.  X  —  0.0915 

IV. 

II. 

V. 

0.2314 

—  2.688 
+  7.010 

+  4-322 

—  0.288 
-8.530 

—  2.445 
+  6.940 

+  4-493 

in.X-5-42 
II. 

II.  +  IV. 

—  S.818 

VI. 

—  I 

+  2.040  =  >- 

—  1.040 

V.  X  — 0.2314 

VII. 

III. 

VIII. 

—    I 

—  0.496 
+  0.496 

+  I. 012 
+  0-053 

—  0.516 

+  o.45> 

-  0.065 

VI.  X  0.496 

III. 

III.  +  VII. 

—    I 

+  1 .065  r=  x 

In  order  to  find  the  values  of  the  unknowns  to  two  places  of  decimals,  the 
computation  should  be  carried  through  to  three  places,  and  the  third  place 
dropped  in  the  final  result. 

88.    Form  {b).     The  logarithmic  solution. 

As  an  example  of  the  logarithmic  method  let  us  take  the  gen- 
eral form  of  the  preceding  example,  when  R  and  6"  are  substi- 
tuted for  the  absolute  terms  0.58  and  8.53  respectively. 

In  the  numerical  work  it  is  better  to  convert  all  the  divisions 
into  multiplications.  Therefore  write  down  the  complementary 
logs,  of  the  divisors  with  the  signs  changed.  Each  multiplier 
may  now  be  written  as  needed  on  a  slip  of  paper  and  carried 
over  each  logarithm  to  be  operated  on.  Thus  for  the  first  oper- 
ation the  slip  would  have  on  it  8.96138  n,  where  the  n  indicates 
that  the  number  is  negative. 

Paper  ruled  into  small  squares,  so  as  to  bring  the  figures  in  the 
same  vertical  columns  and  facilitate  additions  and  subtractions, 
renders  the  work  more  mechanical,  and  is  consequently  an 
assistance  to  the  computer. 

In  general  solutions,  when  the  number  of  unknowns  is  large, 
it  will  be  found  much  better  to  carry  a  double  check,  one  for  the 
coefficients  of  x,  )\  .   .  .  and  the  other  for  the  coefficients  of  R^ 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN 


113 


S,  .  .  .  Though  unnecessary  in  our  example,  it  is  inserted  for 
illustration. 

It  will  be  noticed  that  the  coefficients  of  R,  S  in  the  values  of 
X,  y  follow  the  same  law  of  symmetry  as  the  normal  equations. 
A  little  consideration  will  show  that  this  is  always  the  case. 

Hence,  attending  to  this,  we  may  shorten  the  computation  by 
leaving  out  the  common  terms.  We  have,  therefore,  one  term 
less  to  compute  for  each  unknown,  proceeding  from  the  last  to 
the  first.     The  case  is  precisely  analogous  to  that  of  Art.  j^. 


X 

y 

Check. 

R 

6- 

Check. 

Rkmakks. 

I. 
II. 

III. 

IV. 

V. 
VI. 

II. 

VII. 

VIII. 

IX. 

X. 

XI. 

10.93 

—  s-42 

1.03862 

(8.96138 «) 

—  s-42 
+  7.01 

0.73400  « 
9.69538 

0.42938  « 

-2.688 
+  7.010 

+  4-322 

0.63568 
(9.36432  «) 

—  I 

—  5-5' 

—  1-59 

0.7411S  « 
9-70253 

0.43653  n 

—  2-732 

—  1-590 

—  I 

0 

8.96138 
+  0.091 

9.69538  « 
—  0.496 

—  I 

—  I 

—  I 

0.0 

9.36432 
+  0.231 

9.05970 
+  0.115 

+  I 
+  I 

0 
8.96138  n 

—  0.091 

9.69538 
-1-0.496 

+  •■40 

0.17493 
9-53925  « 

—  0.346 

9.23463  « 

—  0.172 

—  0.091 

log  I. 

III.  —  log  10.93 

Nos. 

IV.-logs.42 

Nos. 
II. 

VI. +  11. 

log.  VII. 

vin.- log  4.322 

Nos. 
Nos.         '" 

—  4-322 

0.63568  « 
0.00000 
+  • 

—  0.496 

0.69548  n 
g.05980 
+  0.11S 

,   8.755'8 
+  0.057 
+  o.ogi 

+  0.148 

XII. 

Xtll. 

—  I 

+  ■ 

+  0...5 

—  0.263 

:.x  =  0.148  R  +  0.1 15  5*, 
J  =  0.1 15  7?  +  0.231  S. 

Substituting  for  R  and  J>'  their  values,  we  have,  as  before, 

X  =  1 .06, 
y  =  2.04. 

89.    Ex.  I.  —  In  the  elimination  of  n  normal  equations  by  the  method  of 

substitution,  show  that  the  total  number  of  independent  coetlicicnls  in  the 

..,,,.,             ,           ..        .;/(«  +  i){n  +5) 
original  and  derived  normal  equations  is ^ • 

[The  sum  is  ^  {1.4  +  2.5  +  .  .  .  «  («+  3)}.] 
Ex.  2.  —  If  the  elimination  of  the  unknowns  in  the  normal  aiuations  is 
carried  out  by  the  method  of  substitution,  the  product 

[aa],  [/>/>.i],  Ur.2]  .  .  . 
has  the  same  value  whatever  order  has  been  followed. 


114  THE    ADJUSTMENT    OF    OBSERVATIONS 

go.  A  method  of  indirect  elimination  by  successive  trials  and 
approximations  has  been  suggested  by  Gauss.  It  will  be  found 
in  Coast  Survey  Report,  1855,  Appendix  44;  von  Freeden,  Die 
Praxis  der Metkode  der  klcinsteii  Quadrate,  p.  96  ;  Vogier,  Aus- 
gleicJiungsrecJinung,  p.  129. 

It  is  not  given,  because  our  experience  has  been  that  it  is  in 
general  intolerably  tedious. 

91.  The  Doolittle  Method  of  Solution.  —  This  method  of 
solution  is  due  to  Mr.  M.  H.  Doolittle  of  the  Computing  Division 
of  the  Coast  and  Geodetic  Survey.*  In  it  there  is  a  combination 
of  improvements  on  the  Gaussian  method  of  substitution.  Its 
advantage  lies  mainly  in  the  arrangement  of  the  work  in  the 
most  convenient  form  for  the  computer.  This  makes  the  solu- 
tion more  rapid  than  by  the  other  method,  the  gain  in  speed  be- 
ing the  more  marked  the  greater  the  number  of  equations. 

In  order  to  make  the  process  employed  readily  followed,  the 
solution  of  the  three  normal  equations, 

\an\x  -h[(/6]y+  lac\z  =  [«/]  , 
\ah\  X  +  [bb]  y  +  [be]  z  =  [bl] , 
[ac]x  +  [bc]y  +  [cc]z=  [r/] , 

is  sriven  in  s:eneral  terms  according  to  this  form. 

The  coefficients  and  absolute  term  of  the  first  equation  arc 
written  in  line  i,  Table  A.  The  reciprocal  of  the  diagonal 
coefficient  [aa]  is  taken  from  a  table  of  reciprocals  and  entered 
in  the  front  column  with  the  minus  sign  prefixed.  The  remain- 
ing terms  of  line  i  are  multiplied  by  this  reciprocal,  and  the 
products  written  in  line  2.  This  gives  x  as  an  explicit  function 
of  J  and  s. 

The  coefficients  and  absolute  term  of  Eq.  2  (omitting  the  co- 
efficient of  x)  are  written  in  line  i.  Table  B.  The  terms  in  line 
2,  Table  A,  beginning  with  that  under  j/,  are  multiplied  by  [ad] , 
the  coefficient  of  y,  and  the  products  set  down  in  line  2,  Table 
B.  The  sum  of  lines  i,  2,  Table  B,  is  now  written  in  line  3, 
Table  A. 

*  See  Appendix  8,  C.  and  G.  Survey  Report  for  1878,  pp.  115-118. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN       115 

Line  4,  Table  A,  is  found  from  line  3  in  the  same  way  as  line 
2  was  found  from  line  i.     This  gives j/ as  an  explicit  function  of  ^r. 

The  coefficients  and  absolute  term  of  Eq.  3  (omitting  the  co- 
efficients of  X  and  j')  are  written  in  line  3,  Table  B.  The  terms 
in  lines  2,  4,  Table  A,  beginning  with  those  under  s,  are  multi- 
plied by  [acl,  [<^r. i],  the  coefficients  of  ^  in  hues  i,  3  respect- 
ively, and  the  products  set  down  in  lines  4,  5,  Table  B.  The 
sum  of  lines  3,  4,  5,  Table  B,  is  written  in  line  5,  Table  A. 

Line  6,  Table  A,  gives  the  value  of  s. 

The  next  step  is  to  find  j  and  x.  The  coefficients  of  the  ex- 
plicit functions  are  written  in  Table  C.  The  absolute  terms  of 
the  explicit  functions  are  written  in  the  first  line  of  Table  D. 
The  value  of  s  is  multiplied  by  the  coefficients  of  s  in  Table  C, 
and  the  products  written  in  the  second  line  of  Table  D.  The 
sum  of  the  numbers  in  column  j'  gives  the  value  of  y  written 
underneath  in  line  3.  The  value  of  j  is  multiplied  by  the  co- 
efficients of  J  in  Table  C,  and  the  products  written  in  the  third 
line  of  Table  D.  The  sum  of  the  numbers  in  column  x  gives  the 
value  of  X. 

92.  The  values  of  x,  y,  z  are  now  found  to  three  places  of 
decimals.  Denote  them  by  x',y',  z' .  If  these  values  are  not 
sufficiently  close,  a  second  approximation  must  be  made.  This 
we  proceed  to  describe. 

First  substitute  the  values  obtained  in  the  original  normal 
equations,  and  carry  out  to  a  sufficient  number  of  decimal  places. 
The  residuals  are  written  in  the  first  line  of  Table  E.  The  co- 
efficients in  line  i,  Table  C,  are  multiplied  by —  [''?^],.  and  the 
products  written  in  line  2,  Table  E.  The  first  recijirocal  in 
Table  A  is  multiplied  by  the  same  residual,  and  the  product 
written  in  column  ,r,  line  i,  Table  F.  The  sum  of  the  numbers 
in  column  2,  Table  E,  is  written  underneath,  as —  \bl.\'\^. 

The  coefficient  in  line  2,  Table  C,  is  multijilied  by  —  [^^/-i], 
and  the  product  written  in  line  3,  Table  E.  The  second  recip- 
rocal in  Table  A  is  multiplied  by  the  same  residual,  and  the 
product    written  in  column  y,  line   i,  Table   Y.     The  sum  of 


ii6 


THE    ADJUSTMENT    OF    OBSERVATIONS 


the  numbers  in  column  3,  Table  E,  is  written  underneath,  as 

The  third  reciprocal  of  Table  A  is  multiplied  by  this  residual, 
and  the  product  written  in  column  z,  line  i,  Table  F.  This 
gives  the  correction  to  the  value  of  s.  The  first  line  of  Table 
F  corresponds  to  the  first  line  of  Table  D,  and  exactly  the  same 
process  employed  in  Table  D  will  complete  Table  F.  The 
total  values  now  are 

x  =  x'  +  x'\ 

y  =  y  +  y  , 
2  =  z'  +  z". 


B 


Recip. 

X 

y 

z 

I 

^\.aa\ 
X  =: 

[ab-\ 
iaa] 

\,aa\ 

I 

\,bb.A 

y  — 

\_bc.^] 
\bb.{\ 

-W-A 
'^\_bbA 

I 

+  [c-..2] 

-VI -A 

id. 7] 

+  VcA 

[a-.2] 

y 

= 

+  \bb-\ 

-Vibe] 

_i_  f"^    r   ^.^ 

\.cc\ 

-[cV] 

D 


y 

z 

[aa] 

[ac] 
[aa] 

[bc.i] 
[bb.i] 

.r 

y 

2 

+  [f£l 

[aa] 

[«^]     , 
[aa]^ 

[bl.x] 

^[bb.A 

[bc.z] 
[ii.i] 

[C/.2] 

'^UC.2] 

z" 

y' 

x' 

INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        117 


X 

y 

z 

-w\ 

-\.cl\ 

-VI. A 

-[iu]i 

X 

y 

z 

2" 

y 

x" 

93.  Addition  of  New  Equations.  —  It  often  happens  that 
after  the  adjustment  of  a  long  series  of  observations,  additional 
observations  are  made  leading  to  additional  equations.  To  make 
a  solution  de  novo  is  necessary,  but  the  work  may  be  very 
materially  shortened  by  the  process  just  given.  Suppose,  for 
simplicity,  that  one  new  condition  has  been  established.  This 
will  give  one  additional  normal  equation  which  may  be  written 
\a(i\  X  +  [bd]  y  +  [cd]  z  +  [dd]  w  =  [d/] ,  ( i) 

w  being  the  new  unknown. 

The  extra  term  to  each  of  the  other  normal  equations  may  be 
written  down  at  sight.     The  complete  equations  are 

[aa]  X  +  [ab]  y  +  [ac]  z  +  [(?</]  a'  =  [<;/] , 

+  [W>]  V  +  [/n]  z  +  \bd'\  w  =  [W] , 

+  [rr]3+[a/]w  =  [r/], 

+  \dd'\w  =  \dl'\. 

Now,  values  of  x,  y,  a  have  been  already  found  from  the  normal 
equations  resulting  from  the  original  condition  equations,  and 
these  values  may  be  taken  as  first  approximations  to  the  values 
of  X,  y,  z  resulting  from  the  above  four  normal  equations.  Sub- 
stitute in  (i),  and 

\iid-\  X  +  [/u/]  /  +  [  r/]  /  +  \,hr\  7C  =  [d/]',  (2) 

where  x',  /,  ^  are  corrections  to  the  approximate  values  of 
X,  y,  z.     The  solution  is  now  finished  as  follows  : 

Form  Table  C  {a)  by  adding  the  extra  column  lo  to  Table  C. 


ii8 


THE    ADJUSTMENT    OP    OBSERVATIONS 


The  term  — i^  ^  ^  is  found  by  multiplying  [acQ  by  the  first  re- 
[aa  ] 

ciprocal.     The  coefficients  of  the  new  equation,  (2),  are  written 

in  the  first  line  of  Table  G.     Since  corrections  to  values  already 

found  are  required,  the  method  of  proceeding  must  be  similar  to 

that  employed  in  Table  E.     The  notation  in  Tables  C  (a)  and  G 

explains  this. 

The  reciprocal  of  the  sum  of  column  za,  that  is,  1^.3],  in 

Table  G  is  written  last  in  the  column  of  reciprocals  of  Table  C 

(a)  with  the  minus  sign.     The  product  of  this  reciprocal  and  the 

absolute  term  —  [d/]'  of  the  new  equation,  that  is,  r  ,,    y  is  an 

approximate  value  of  zv.  This  value  of  w  is  multiplied  by  the 
terms  in  the  last  column  of  Table  C  (a),  and  the  products  are 
written  in  the  first  line  of  Table  H.  Column  s  gives  the  cor- 
rection to  2.  Table  H  is  now  completed  in  the  same  way  as 
Tables  D  and  F. 

C  (a)  G 


y 

z 

IV 

I 

lab\ 

Wc\ 

[ad] 

[aa] 

\,aa-\ 

\_ad\ 

[aa] 

I 

T 

lbc..-\ 
[M.i] 

[M.<] 
[M.i] 

[Cd.2] 

ICC.2] 

[CC.2] 

I 

['i'f-3] 

X 

y 

z 

■w 

[a.r\ 

[bd] 

[aa\ 

led] 

^;''\  lad] 

VbbA  ^^''•'1 

\.dd\ 

ibd.i] 

[a/.2] 

idd.-i] 

H 


X 

1 

IV 

iaa\ 
laaY 

ibb.A'" 

\hc.x]    , 

Vbb.A  " 

\cd.A 

—  t -w 

\CC.2\ 

,  \dn' 

z' 

w 

y' 

x' 

INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        119 

94.  In  order  to  illustrate  this  method  still  further,  the  solu- 
tion of  the  following  equations  is  shown  in  the  form  indicated  on 
the  preceding  pages. 

Suppose  given  the  normal  equations  : 

1.  o  =  -H  5.4237  w  +  2.1842 .V—    4.3856  v+ 2.3542s— 3.6584, 

2.  o  =  +  2.1842  w  +  6.9241  .V  —  1.2 1 30s  +  2.8563, 

3.  0=—  4.3856  Z£^  +   I  2. 8242  T+  3. 4695  s  +  8. 742  I, 

4.  o  =+ 2.3542  w  —  1.2  130  .V  +     3.4695  ;V+ 7.12433  + 0.6847. 
The  solution  is  conducted  as  follows  : 


A 

B 

I 

I 

2 
3 
4 

2 

3 

4 

5 

6 

7 

8 

I 

1 
2 
3 
4 

5 

6' 
7 

8 
9 

2 

3 

4 

S 

6 

I 
2 

S 

6 

-."184 
-.'165 

■zy 

+3-424 
•zv  =: 

X 

+  2.184 
—    .401 
+  6.048 

X  ^^ 

y 

-  4-386 
+  -807 

+  >-759 

—  -29 

z 

+  2-354 

—  -433 

—  2.157 
+     -356 

-3.658 
+     -6731 
+  4-323 

—    -7'33 

3 
4 

7 
8 
9 

12 
13 
14 

•5 

+€ 

X 

•9 
.8 

76 

y 
+ 12 

759 

82 

54 

5' 

2 

-1.213 

—  -944 
+  3-470 
+  1.900 
+    -626 

+  7.124 

—  1. 019 

—  .768 

—  4.101 

+  2.856 
+  1-467 
+  8.742 

—  2.950 

—  >-254 

+  0.685 
+  «.584 
+  >-539 

—  3.104 

S 
6 

10 
1 1 

—.114 

+  8-77 

y  — 

+  5.9961+4.538 
-     .684    -    .5173 

7 
8 

16 

'7 

+  1.2^6     -l-0.70i 

—.809 

—    -57 

The  first  column  in  each  of  the  above  tables  gives  the  number 
of  the  line,  and  the  second  the  order  of  procedure. 

It  is  to  be  observed  that  the  numbers  in  Table  B  have  but  a 
single  use,  while  those  of  Table  A  are  used  over  and  over  ;  and 
when  the  number  of  equations  is  large,  it  is  of  great  advantage 
that  they  should  be  thus  tabulated  by  themselves  in  a  form  com- 
pact and  easy  of  reference. 


c 

II 

Reciprocal. 

X 

y 

z 

TO 

X 

y 

= 

I 

2 

3 

4 

-..84 
-.,65 

—  .114 

—  .809 

—  .401 

+  .807 
—  .29 

—  •433 

+  -356 

—  .6.S4 

I 

2 
3 
4 

+   •673" 
+  .2468 

.1024 

+  -3524 

—  •7'33 

—  .2029 
+  .0368 

—  .879^ 

—  •S'73 
+  .3899 

=;r  1 

—y\ 

—  '1 

—  .127 

+   ..I7 

' 

THE  ADJUSTMENT  OF  OBSERVATIONS 


E 

F 

I 

2 

3 

4 

w 

JC 

J' 

z 

2 

3 
4 

—  .0165 

+  .0169 
+  .0066 

+  -0047 

—  -0133 

—  .0068 

+  -0039 
+  .007 1 
+  .0084 
+  .0,02 

I 
2 
3 
4 

+  .0030 
+  .0103 
+  .0146 
+  .0071 

—  .0039 

- .0085 

—  .0052 

—  .0176 

+  .0018 

+  .0.63 

—  0239 1 

—  32     ' 

—  ^2 

—  ^■2 

=  7^2 

+  -0235 

+  .0.8. 

—  -0154 

+  .0296 

+  -0353 

As  the  multiplication  is  performed  by  Crelle's  Tables,  no  mul- 
tiplier is  allowed  to  extend  beyond  three  significant  figures. 
Other  numbers  may  be  extended  to  four  ;  but  it  would  be  a 
waste  of  time  to  extend  any  number  farther,  except  in  the  pro- 
cess of  substitution  for  the  determination  of  residuals. 

If  an  arithmometer  is  used,  four  significant  figures  may  be  re- 
tained throughout.  It  will  seldom  be  advisable  to  retain  more 
significant  figures  than  this. 

For  the  sake  of  perspicuity  in  explanation  and  convenience  in 
printing,  we  have  here  made  some  slight  departures  from  actual 
practice.  For  instance,  in  the  solution  of  a  large  number  of 
ecjuations,  it  would  be  inconvenient  to  pass  the  eye  and  hand  out 
to  a  vertical  column  of  reciprocals  ;  and  they  are  better  written 
in  an  oblique  line  near  the  quantities  from  which  they  are  de- 
rived and  with  which  they  are  to  be  employed. 

By  this  process,  Mr.  Doolittle  solved  in  five  and  one-half  days, 
or  36  working  hours,  with  far  greater  than  requisite  accuracy, 
41  equations  containing  174  side  coefficients  counting  each  but 
once,  or  430  terms  in  all. 

95.  Suppose  that  after  the  solution  of  the  foregoing  equations 
and  the  consequent  adjustment  a  new  condition  is  established, 
resulting  in  the  following  normal  equation  :  o  =  —  2.0475  ^v  -f 
0.8362  X  -f  1.8567  J  —  1.3 14Q  -'^  +  8.2527  //  —  1.8372  ;  with 
the  addition  of  the  term  —  2.0475  ^'  to  the  first  of  the  previous 
equations,  +0.8362  ?i  to  the  second,  etc.  The  al^solute  term 
1.8372  is  supposed  not  to  be  an  original  discrepancy,  but  an  out- 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN'        121 

Standing  residual,  after  the  foregoing  solution  has  fully  entered 
into  the  adjustment,  as  is  generally  the  case  with  azimuth  and 
length  equations,  in  a  triangulation  adjustment. 


C(<M 

('. 

Recip- 
rocal. 

.r 

y 

2 

u 

•w 

X 

y 

z 

u 

I 

2 
3 
4 
5 

-..84 
-.165 

—  .114 

—  .809 

—  -145 

—  .401 

+  .807 
—  .29 

—  •433 
+  •356 

—  .684 

+  •377 

—  .274 
+  -0307 

—  .282 

I 
2 
3 
4 

5 

—  2.05 

+  .836 
+  .822 

+ 1.851 
— 1.654 
-  .481 

—  1-3IS 

+    .888 
+    -59' 
+    -184 

+  8.253 

—  -773 

—  -455 

—  .008 

—  .098 
+  6.9,9 

+  .166 

-  .269 

+    .348 

H 

7V 

X 

y 

z 

u 

I 
2 
3 
4 
5 

+  .1007 
+ .0326 
+  .0482 
+ .0469 
+  .228 

—  0732 

—  .0268 

—  •0173 

+  .0082 
+  -0515 

—  -0753  ( 

+  -267  1 
=  «1   ' 

—  yz 

+  -0597 

—  .117 

The  Precision  of  the  Most  Probable  {Adjusted)    Values. 

96.  The  problem  now  before  us  is  to  find  the  p.  e.  of  the  un- 
knowns, X,  y,  .  .  .  dcs,  determined  from  a  scries  of  normal  equa- 
tions. If  the  observation  equations  are  reduced  to  the  same 
unit  of  weight,  which  we  shall  take  to  be  unity  for  convenience, 
the  general  form  of  the  normal  equations  is 


[aa]x  +  [ah']y  +  -  ■  ■  =  [<//], 
[ah'\x  +  [hh']y+  ■  •  •  =  [hi]. 


(I) 


Let 


;-  =  the  p.  e.  of  a  single  observation. 
^xi  ^yi  '  •   •   =  the  p.  e.  of  X,  y,  .  .  . 
px,  py,-  ■  •   =  the  weights  of  x,  y,   .  .  . 

From  Art.  47  we  have 


122  THE    ADJUSTMENT    OF    OBSERVATIONS 

Pxr^^  =  PyTi  =  •    •    •  =  7-2.  (2) 

In  order,  therefore,  to  determine  r^.,  r^,  ...  we  must  make 
two  computations,  one  of  the  weights  p^.,  py,  .  .  .  and  the  other 
of  r,  the  p.  e.  of  a  single  observation. 

It  is  evident  from  an    inspection    of   the    normal    equations 
that  X,  y,  .  .  .  are  linear  functions    of    /^,  l^,  .   .  .     Let,  then, 

X  =  aj^   +  a.J^  +  •    •  •   +  aj„  =  [a/] , 

y = {s,k  +  pj,  +  •  •  ■  +  /?«/» =  im .  (3) 


in  which  a,,  a,,,  ...;  ^^,  ^,,  ...;..  .  are  functions  of  «,,  d^, 

.  .   .  ;  a.„  d.„  ...;..  .  their  values  being  as  yet  undetermined. 

Now,  r  being  the  p.  e.  of  each  of  the  observed  quantities  M^, 

M~2,  ■  ■   ■  M,„  must  be  also  the  p.  e.  of  /,,  4  .   .  .  4,  which  differ 

from    Ml,    M,,,   .  .  .  M„     by   known    amounts    (see    Art.    74). 

Hence,    since    /,,   /„,   .  .  •  /„    are    independent    of    each    other 

(Art.  13), 

rj=r[aa],  ;-/  =  r[/3^],.   .  .  (4) 

and  therefore 

We  shall  first  of  all  determine  the  weights  p,.,  py,  .  .  . 

97.  The  demonstration  may  be  carried  out  simply  by  the  ap- 
plication of  the  principles  of  undetermined  coefficients.  Thus, 
substitute  [a/],  [13/],  .  .  .  iox  x,  y,  ...  in  the  normal  equa- 
tions (i),  and 

[aa]  [a/]  +  [ah]  [^8/]  +  •  •  ■  =  [«/], 

[ah]  [a/]  +  [hb]  [(il]  +  .  .  •  =  [hi],  (6) 


or,  arranging  according  to  l^,  l^,  .  .  . 

{[aa]a,  +  [ah](i,  +  - a,]l, 

+  {[««]  a^  +  [ah]  P,  +  -  •  •  -  ^2)^2  + 
{[ab]a,  +  [hh]l3,  +  -  ■  --h,}/, 

+  {[ab]  a,  +  [bh]  fi,  +  -  ■  ■-b,}l,+ 


=  o. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        123 

The  unknown  quantities  a,,  a„  •  •  •  may  be  so  determined  that 
the  coefficients  of  /j,  /.„  .  .  .  shall  each  equal  zero.  Hence  the 
several  sets  of  equations, 

r  [(/(/]  a^  +  [dh]  /^i  +  •  •  ■  —  ai  =  o, 
\  [aa]  tt.,  +  [ah]  ^.,  +  •  ■  •  -  a..  =  o, 
'./...■....■..  (7) 

[ah]  a,  +  [hh]  /3^  +  .   .   .  -  /,j  =  o, 

[ab]a,  +  [bb]l3,  +  -  ■  .  -b,  =  o, 


1 


are  simultaneously  satisfied  by  the  same  values  of 

Multiply  the  equations  of  each  set  by  a^,  a.^,  .   .  .  in  order, 
and  add  ;  then  necessarily 

[aa]=i,[af3]  =  o,[ay]  =  o,   ...  (8) 

In  a  similar  way,  multiplying  by  ^j,  d.^,  .  .  .  -y  c^,  c^y  .  .  .  ,  etc., 
and  adding,  there  result 

[baj^o,  [bj3]=  I,  [by]  =  o,  .  .  . 
[ca]  =  o,  [r^]  =  0,  [ry]  =1,  .   .    . 


Again,  multiply  the  first  set  by  a^  a^,  •  •  •  ,  the  second  by 
yS,,  y8^,  .  .  .  ,  and  so  on,  and  add,  and  we  have  the  sets  of  equa- 
tions, 

\  [aa]  [aa]  +  [ab]  [a(3]  +  ■  •  •  =  [,7a]  =  I, 
i  [ab]  [aa]  +  [bb]  [tt/3]  +  .  .  .  =  [ba]  =0, 
I (9) 

f  [aa]  [afi]  +  [ab]  [/3/3]  +  •  •  •  =  [afi]  =  o, 
^[ai][a(3]  +  [bb][l3(3]  +  ..  ■=[bft]=  I, 


from  which  equations  [aa].  [a/3],  .  .  •  may  be  found. 

It  is  plain  that  the  coefficients  of   [aa],  [a^].  •  •  •  ;   [a^], 
[/3/8],   ...   in   these  equations  arc  the  same  as  tliosc  of  x,  y. 


124 


THE    ADJUSTMENT    OF    OBSERVATIONS 


...  in  the  normal  equations,  and  that  the  absolute  terms 
are  i,  o,  .  .  .  ;  o,  i,  ...;..  .  instead  of  [^?/],  [<^/],  .  .  . 
Hence, 

98.  To  Find  the  Weights  of  the  Unknowns.  —  In  the  first 
no  una/  equation  write  i  for  [^?/],  and  in  the  other  normal  equa- 
tions put  o  for  each  of  [/^/] ,  [r/] ,  .  .  .  ;  the  value  of  x  found 
from  these  equations  zvill  be  the  reciprocal  of  tJie  tveight  of  x,  and 
the  values  of  y,  z,  .  .  .  Zi'ill  be  the  values  of  [a^],  [a7],  •  •  • 
1)1  the  second  normal  equation  ivrite  \  for  \bl\  and  in  the  other 
equations  put  o  for  each  of  \al^^,  \cl'\,  .  .  .;  the  value  of  yfojind 
from  these  equations  zvill  be  the  reciprocal  of  the  zveight  of  y,  and 
the  values  of  z,  .  .  .  found  zvill  be  the  values  of  [/S7],  .  .  . 
Similarly  for  each  of  the  unknowns  in  succession.  For  example, 
the  weight  equations  for  three  unknowns  are 


[aa] 

[«^] 

[avJ 

[a^] 

im 

[Py] 

Wyl 

[Pv] 

[yv] 

0 

0 

+  \,aa\ 

IK] 

+  m 

I 
0 
0 

+  ["-] 

0 

0 

-\-\ab\ 

+  ["■] 

The  quantities  [a/3],  [^7],  •  •  •  are  necessary  when  the  weight 
of  a  linear  function  of  the  unknowns  is  required,  as  will  be  seen 
presently.     (See  Art.  108.) 

99.  It  is  evident  from  the  form  of  the  weight  equations  that 
if  the  elimination  is  carried  through  by  the  method  of  substitu- 
tion, the  successive  steps  to  the  left  of  the  sign  of  equahty  are 
the  same  as  in  Arts.  86-88.  Hence  if  the  equations  are 
arranged  so  that  the  unknown  whose  weight  is  required  —  z,  for 
example  —  is  found  first,  we  should  have  the  forms 


X 

y 

z 

[ay] 

[Sy] 

[yy] 

\aa-\ 

-^\.bb.C\ 

+  /....] 

+  [CC.2. 

\bl.x\ 
[cb.2] 

[aa] 

+  [ab] 
+  [M.,] 

+  [ac] 

0 
0 
I 

.-.  rrf.2]  =  =[^/.2] 

.-.  [cc.2'\  [vv]  =  I 

INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        125 

Hence  the  coefficient  of  the  unknown  first  found  in  the  ordinary- 
solution  of  the  normal  equations  is  the  weight  of  that  unknown. 
By  a  separate  elimination  for  each  unknown,  the  weight  of  that 
unknown  could  be  found  as  above,  but  the  process  would  be  in- 
tolerably tedious. 

100.  Special  Cases  of  Tzuo  and  Three  Uiikno^vus.  —  We  may, 
however,  from  the  preceding  derive  formulas  for  the  weights  in 
a  series  of  normal  equations  containing  not  more  than  three  un- 
knowns, which  are  easy  of  application. 

Thus  wdth  two  unknowns,  x  and  j,  y  being  found  first, 


Py 

=  [bh.l1 
=  m  - 

[abf 
[aa] 

In  the 

reverse 

order. 

X 

being  found  first. 

p^ 

=  L^'^]  - 

[abf 

or 

_    X  _  J^ 

^^~[bby    ^"~[aa]' 
where 

A  =  [aa]  [bh]  -  [ab]  [ab]. 

With  three  unknowns,  x,  y,  .v,  performing  the  elimination  of 
the  normal  equations  in  the  order  ^,  y,  x,  we  have 

p,  =  [cc.2]  , 

[bb.i-\[cc.2-\ 


Py  = 

Px  = 


[rr.i] 
[aa][bb.i][cc.2] 


[bb]  [re] -[be]  [be]' 
which  expressions  are  easily  transformed  into 

X 

^^  ~  [aa]  [bb]  -  [abY  ' 

_  A. 

^"  ~  [aa]  [cc]  -  [acY  ' 


[26  THE    ADJUSTMENT    OF    OBSERVATIONS 


where 

X  =  [aa]  [hb]  [cc]  +  2  [ab]  [be]  [ac]  -  [aa]  [bcf  -  [bb]  [acp  -  [cc]  [abf. 

From  these  formulas  the  weights  of  the  unknowns  can  be  found 
directly  without  solving  the  normal  equations.  If  the  normal 
equations  have  simple  coefficients,  it  is  much  more  rapid  to  find 
the  weights  in  this  way  and  solve  the  equations  by  ordinary 
algebra  rather  than  by  the  Gaussian  method.  But  when  the 
number  of  unknowns  exceeds  three,  this  becomes  too  cumber- 
some. 

Ex.  —  To  find  the  weights  of  the  adjusted  angles  in  Ex.  4,  Art.  77. 

Here 

X  =  12  X  II   X  15  —  I2X  16  —    II   X  49 

=  1249 
and 

A  =  ^9  ^8.4, 
149 

,      1249 

A  =^  =  9-5, 

A  =  ^  =  95. 

If  Ux,  Uy,  Ui  denote  the  reciprocals  of  A^A)  A^'^spectively,  then 

2lx  =  O.  II93, 

Uy     =     0.1049, 

Itz    =   0.1057. 

lOI.  Modification  of  Geiieral  MetJiod. — To  carry  out  the 
method  of  Art.  98  directly  as  stated  would  be  excessively 
troublesome,  and  various  modifications  have  been  proposed. 
The  following  scheme,  which  consists  in  running  the  weight 
equations  together,  will  be  found  very  convenient. 

Take,  for  simplicity  in  writing,  three  unknowns,  x,  y,  z,  and 
to  the  ordinary  form  of  the  normal  equations  as  arranged  for 
solution  add  the  columns 

100 

010 

001 
the  check  being  carried  throughout. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN 


1-^7 


Perform  the  elimination  exactly  as  stated  in  Art.  86,  and  find 
the  values  of  the  unknowns  in  the  usual  \va}-.     We  have  then 


where 


[aa]       [bb.i] 
[bb.i] 


+  5, 


.r 

y 

= 

R 

S 

r 

Clieck. 

\aa] 

{^] 

w^ 

\,al\ 

C      I 

„ 

o 

["•']  +  > 

i"^] 

'M' 

lbc\ 

Vbl] 

o 

I 

o 

/■^    +  I 

[ac] 

'bc\ 

Ice] 

Vd] 

1        ° 

o 

I 

[./  +  . 

{ab^ 

Wc] 

["/] 

I 

aa\ 

\_aa-\ 

[aa\ 

[""] 

f 

■*<5..] 

ibc.A 

[W.i] 

r   ^t 

' 

\bc.i\ 

ibl.^] 

[bb.,] 

■i 

1 

I 

[<rc-.2] 

[r/.2] 

r  -5"2 

/"'" 

{c!.A 

\      R. 

^  [CC.2] 

1   ■ 

I 

UcA 

I    [cc.^■\ 

^  [cr.2] 

' 

Now,  taking  the  first  column  in  the  table  under  the  heading 
R,  and  attending  to  Art.  98,  we  have 


[ay]  = 


R. 

[r..2]' 


[«y] 


[W.i]       [rc.2]' 


I  [«6],__^,        [</r] 


"-  =  '^'"^  =  M"M^"^^ 


^  +.^+ 


Similarly  ff)r  the  column  under  S, 


M 


128  THE    ADJUSTMENT    OF    OBSERVATIONS 


M  = 


[CC.2]  ' 


tty  =  im  =  f^-r^  + 


[W.I  J  [cC.2] 

and  for  the  column  under  T. 


I'z  =  [yy]  = 


Also  it  is  evident  that 


[cC.2] 

[cC.2] 

^,_[bl.l]  [Cl.2] 

M  +  CM^+M:!]^ 


The  forms  of  the  expressions  for  [aal,  [/3^],  [77],  .  .  . 
show  that  these  quantities  may  be  conveniently  computed  from 
the  preceding  tabular  elimination  scheme.  Thus  the  sum  of  the 
products  of  each  pair  of  numbers  bracketed  under  the  heads 
J^,  S,  T  will  give  ;/,.,  u,j,  ti^  respectively. 

The  convenience  of  this  form  is  seen  in  such  a  case  as  the 
following,  which  is  of  common  occurrence.  In  a  set  of,  say,  40 
normal  equations,  the  weights  of  10  of  the  unknowns  may  be  re- 
quired. These  10  would  be  placed  last  in  the  solution  of  the 
equations,  and  the  extra  columns  R,  S,  .  .  .  added  after  30  of 
the  unknowns  had  been  eliminated,  thus  giving  the  weights  re- 
quired, with  a  trifling  increase  of  work. 

102.  Ex.  I.  —  Given  the  normal  equations 

\2  X  —    T  z  =  R 

-\-  \\  y  —    A.  z  =  S 
-    7  X  -    4j/  +  i5^=7' 
to  find  the  weights  oiy  and  z. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        129 


X 

y 

z 

S 

T 

+  12 

0 
+  II 

-  7 

-  4 

-  7 

-  4 

+  15 

I 

0 

-  0.5S33 

4-  II 

-  4 

+  0.0909  ) 

-  4 

+  10.9169 

+  I 

I 

-  0.3636 

+  9-4625 

I 

+  0.3636  ( 
+  0.0384  \ 

+  0.1057  ( 

Hence 

Uz=  \  Y.  0.1057  =  0.1057, 

7/y  =  I  X  0.0909  +  0.3636  X  0.0384  =  0.1049, 
agreeing  with  the  values  in  Art.  100. 
Ex.  2.  — Show  that 

{bl-  I]  =  [«/]  7?i  +  \bl\ 

id.  2]  =  \al\R.,  +  [^/]  6-,  +  [r/], 


Ex.  3.  — Show  that  the  multipliers  7?„  7?„  .  .  .  satisfy  the  conditions 
\ad\  /?,  +  \ab'\  =  o, 
\_aa\  R.,  +  lab\  S,  +  [ac]  =  o, 
[ab  ]  R,  +  [bb  ]  S,  +  [be]  =  o, 
[aa]  R.,  +  [ab]  S^  +  [ac]  T^  +  [«^]  =  o, 
[ab]R,  +  [bb]  S,  +  [be]  T,  +  [bd]   =  o, 
[ac  ]  /?3  +  [be  ]  S,  +  [ec]  T,  +  [cd]  =  o, 


103.  Second  Method  of  Finding  the  Weights  of  the  Un- 
knowns.—  If  we  multiply  the  first  of  the  normal  equations  i. 
Art.  96,  by  [aal,  the  second  by  [a/S],  the  third  by  [a7],  and 
so  on  ;  add  the  products,  and  attend  to  equations  9,  Art.  97,  we 
obtain 


Similarly 


X  =  [ou]  [a/]  +  \a/3]  [hi]  +  [ay]  [cl]  + 

2  =  MM  +  Wy]W\  +  [yy]['-/]  + 


I30  THE    ADJUSTMENT    OF    OBSERVATIONS 

Hence,  since  [aa],  [/S/S],  .  .  .  are  the  reciprocals  of  the 
weights  of  X,  y,  .  .  . ,  this  method  of  finding  the  weights  may 
be  stated  as  follows  : 

/;/  any  given  scries  of  observations,  having  formed  the  normal 
equations,  replace  the  ninnerical  absolute  terms  by  the  general 
symbols  \al^,  \pl^,  •  •  •  and  find  by  any  method  of  elimination 
the  values  of  X,  y,  .  .  .,  in  terms  of  [^/],  l^bl'^,  .  .  .;  then  the 
weight  of  X  is  the  reciprocal  of  the  coefficient  of  [«/]  in  the  vahie 
of  X,  the  zveight  of  y  is  the  reciprocal  of  the  coefficient  of  [bl'\  in 
the  value  of  y,  and  so  on. 

The  coefficients  of  the  remaining  symbols  for  the  absolute 
terms  in  the  expressions  for  x,  y,  .  .  .  give  the  values  of  [a/3], 
[a7],  .  .  .  ;  [/^y],  ...;...  and  the  numerical  values  of  the 
unknowns  x,  y,  .  .  .  may  be  found  by  substituting  for  [<^/J, 
[^/],  .   .  .  their  numerical  values. 

In  this  method  of  computing  the  values  of  the  unknowns  and 
their  weights,  a  machine  can  be  used  with  great  advantage. 

The  formulas  of  Art.  loi  are  easily  derived  from  the  preced- 
ing principles.     For  solving  the  normal  equations 

[aa\  X  +  [ab]  y  +  [ac]  z  =  \al\ 
{ah\  X  +  {hh\  y  +  [/)rl  z  =  \bl\, 
\ac\  X  +  \[k\  y  +  \cc\  z  =  [cl], 

by  the  method  of  substitution  we  have  for  the  first  unknown, 
Comparing  this  with  the  general  expression  for  x  in  Eq.  i. 


=  '"'ife 


{aa\       \bb.i\       [cc.2] 
r    1        ^^ 


INDIRECT    OBSERVATIOXS    OF    ONE    UXKXOWN        131 

Similarly  for  j'  and  c. 

Ex.  —  To  find  the  weights  of  the  unknowns  in  Ex.  4,  Art.  77. 
Solving  the  normal  equation.s  in  general  terms, 

X  =  0.1 193  [a/]  +  0.0224  [/;/]  +  0.0616  [^/], 
y  =-  0.0224  [a/]  +  0.1049  [/?/]  +  0.0384  [t7], 
2  =  0.0616  [a/]  +  0.0384  [d/]  +  0.1057  [cl]. 

Hence  «»  =  [««]  =  0.1193, 

«y  =  W]  =0.1049, 

»z  =  [77]  =  0.1057, 
as  found  in  Art.  100. 

104.  In  deducing  the  formulas  for  the  precision  of  the  ad- 
justed values  in  a  series  of  normal  equations,  we  have,  for  con- 
venience in  writing,  taken  the  observation  equations  to  be 
reduced  to  weight  unity,  and  the  normal  equations,  consequently, 
to  be  of  the  form 

[aa]  X  +  [ab]  y -\-   .  ■  .  =  [al], 

[ah]  X -{- [hh]  y  +   ■  .  .  =[h/],  (i) 


The  formulas  with  the  weight  symbols  introduced,  corre- 
sponding to  those  found  in  the  preceding  articles,  are  easily 
derived  from  them  by  writing  a  V/,  b  ^p,  .  .  .  I  V/,  for  a, 
b,  .  .  .  I,  and  a-\lu,  ^  V//,  ...  for  a,  /3,  ...  respectively. 
(See  Art.  48.) 

Thu.s,  for  example,  from  the  normal  equations 

[pad\  X  +  [pah]  v -\-   ■  ■  ■  =  [pal], 

[pah]  X  +  [phh]  V  +  .  .  .  =  [phi],  (2) 


we  should  have 

X  =  [al]  =  [iimi]  [pal]  +  [ua^]  [phi]  -f   •  •  • 

y  =  [jil]  =  [uafi]  [pal]  +  [um  [phi]  +    .  •  .  (3) 


and  by  equating  coefficients  of  /,,  /.,,...   in  the  first  expression, 

[uaja]  a  I  -I-  [iia/S]  hi  +   •  •  •   =  /^,ai, 

[ua(S]a,  +  [am'>i+   ■  ■  ■   ="i(^v  (4) 


132  THE    ADJUSTMENT    OF    OBSERVATIONS 

105.    To  Find  the  p.  e.  r  of  a  Single  Observation.  —  If  the 

errors  A  were  known  —  that  is,  if  the  n  observation  equations 
were 

^1^0  +  -^iJo  +    •  •  •    -  A  =  ^1. 

a^x^  +  b.,y^  +  ...  -  /,,  =  A,„  (i) 


where  .\\^^  y^  .  .  .  are    the  true  vahies  of  the  unknowns  —  we 
should  have  at  once 


71 


But  we  have  only  the  residuals  7'  with  the  observation  equations 
a^x  4- 1)^}'  +  •  •  •  —  /i  =  z\, 
a^x  +  b.,y  +  ...  —  /,  =  Vnj  (2) 


where  x,  y,  .  .  .  are  the  most  probable  values  of  the  unknowns. 
We  must,  therefore,  express  [A'-']  in  terms  of  the  residuals  v  in 
order  to  find  /a. 

From  the  two  sets  of  equations,  by  subtracting  in  pairs, 

A^  =  z/j  +  a^  {x^  —  x)  +  b^  (_t'o  -  J')  +  •  •  • 

A2  =  z/,  +  a,  (^0  -  x)  +  b^_  (j'o  -  J')  +   •  •  •  (3) 


Now,  taking  the  m.  s.  e.  /*,.,  /u.,^,  ...  to  be  the  errors  of  x,  y, 
.  .  .  ,  that  is,  to  be  equal  to  x^  —  x,  y^  —  y,  .  .  .  ,  we  have  from 
Eq.  4,  Art.  96, 

Xo-x  =  IX  Vi^,    yo-y  =  H-  V[/3^J,  .  .  . 
and  therefore 

A,  =  V,  -[-fx(a,  vh  +  b,  ^im  +...). 

A,  =  z/,  +  /x  (a,  \^  4-  ^2  \/[(3(3]  +...)• 


Squaring,  adding,  and  attending  to  equations  (9),  Art.  97,  we  have 
approximately,  w^  being  the  number  of  unknowns, 

[A^]  =  [zr~]  +  ;;,;x^  (4) 

Putting  [A-]  =  ;/yu.-,  there  results 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        133 


-M_  and  ,=. 674s  VZ-J^. 


ix'  =  ,^-::7^  and  r  =  .6745  \/  -^-^  ,  (5) 

the  expression  required. 

Reasoning  as  in  Peters'  formula,  Art,  32,  we  easily  deduce 
from  (4) 

r=  0.8653 -JL=,  (6) 

V«  (//  —  «,) 

which  is  known  as  Liiroth's  formula  {Astron.  NacJir.,  1740). 

When  «,  =  I,  equations  (5)  and  (6)  reduce  to  Bessel's  and 
Peters'  formulas  respectively  (Arts.  29,  32). 

106.  Methods  of  Computing  [t^"].  —  {a)  The  ordinary  method 
is  to  substitute  the  values  of  the  unknowns  found  from  the  solu- 
tion of  the  normal  equations  in  the  observation  equations,  and 
thence  find  v^,  v.„  .  .  .  The  sum  of  the  squares  of  these  resid- 
uals will  give  [z^] . 

The  residuals  having  to  be  found,  for  the  purpose  of  testing 
the  quality  of  the  work  this  method  of  computing  [7'"]  is  on  the 
whole  as  short  as  any. 

As  checks  on  the  values  of  [t/]  found  in  this  way  the  follow- 
ing are  of  value  : 

(d)  If  we  multiply  each  observation  equation  by  its  2'  and 
take  the  sum  of  the  products,  then,  remembering  that  [^av']  =  o, 
l^t'l  =  o,  .  .  .  ,  we  find 

(c)  If  we  multiply  each  of  the  observation  equations  by  its  / 
and  take  the  sum  of  the  products, 

[a/]  X  +  [/>/]  y-\-  .  .  .  -\//]  =  \7-/]  =  -  [tfi]. 

{d)    We  have  for  two  unknowns,  x  and  j', 

[7.2]  =  [(ax  +  by  -  iy\ 

=  [an]  x'  +  2  \al>]  xy  +  \l>f>]  v-  -  2  [at]  x  -  2  \t>l]y  -(-  [//] 


134  THE    ADJUSTMENT    OF    OBSERVATIONS 

-(.i«]-I^m).+[//j-|::^ki 

and  generally  for  m  unknowns, 

[,,2]  =  [(ax  +  hy+  ■  ■  ■  -  ly] 

=  [«a]    .V  +  j^  V  +  •  •  ■  -  ^ 


Now,  from  (9),  Art.  85,  the  coefficients  of  [aa^,  [dd.i^,  .  .  .  are 
each  equal  to  zero.     Hence 

[vv]  =  [11.  m] 

[alf       [hi. if       [cl.2f 


=  [//] 


[aa]       [bb.i]        [cc.2'] 


This  expression  was  first  given  by  Gauss  (De  Elementis 
Ellipticis  Palladis,  Art.  13).  Its  form  suggests  that  if  we  add 
an  extra  column  to  the  normal  equations,  as  shown  in  the  follow- 
ing scheme,  we  shall  find  [7'^]  at  the  same  time  as  the  first  un- 
known. This  is  analytically  very  elegant,  and,  as  the  check 
(see  Art.  85)  can  be  carried  with  this  column  through  the  solu- 
tion of  the  normal  equations,  it  may  be  used  for  finding  [7'^],  if 
one  is  computing  alone.  Only  one  extra  term  [//]  has  to  be 
computed  while  forming  the  normal  equations. 

The  scheme  is  as  follows  : 


INDIRECT    OBSERVATIOXS    OP    ONE    UNKNOWN        135 


X 

y 

- 

[aa] 

+  [ai] 

+  [ac] 
+  [dc] 
+  [cc] 

[al] 
[bl] 
[cl\ 
[11] 

I 

[ai] 

-^  [aa] 

+  [ii.i] 

^[ac] 

[aa] 

+  [fie.  I] 

+  [cc.x] 

lal] 
[aa] 
[bl.\] 
[cU] 

[//.x]-[//]       [fj[./] 

I 

Abc.x] 

^  [bb.^] 

[cc.z] 

[blA] 
[bb.\] 
[6-/.  2  J 

107-  Ex.  I. — To  find  the  m.  s.  e.  of  the  adjusted  values  of  the  un- 
knowns found  in  Ex.  4,  Art.  77. 

The  first  step  is  to  find  [i)n'~\.  This  we  .shall  do  in  the  four  ways  in- 
dicated. 


(a) 


V 

p 

PrP- 

—  0.05 

5 

O.OI 

-0.36 

7 

0.91 

+  0.68 

4 

1.85 

—  0.03 

7 

O.OI 

-0.62 

4 

1-54 

4.32  =  [pir] 

(^) 


Pxh'Vx  =  0 

pMv^  =  0 

P:l.(i>;  =  0 

P^l^'^U  =  7  X  0.76  X 

—  0.03  =   —  0.16 

A4^5  =  4  X    1.66  X 

—  0.62  =   —  4.12 

4.28  =  +  [/^7'/i  =  -r/"''i 


136 


THE    ADJUSTMENT    OF    OBSERVATIONS 


(4 


\_pal'\  X  =  —    5.32  X  —  0.05  =  0.27 

pbl  \  y  =  —    6.64  X  —  0.36  =  2.39 

\j)cl'\  z  =  -\-  11.96  X  +  0.68  =  8.13 

10.79 

Pxhh  =  0 

pJJ.  =  0 

Pzk^i  =  0 

AAA  =  7  X  (0.76)2             =   4.04 

A44  =  4  X  (1.66)'            =  11.02 

15.06 

4.27  = 

=  ipv'-\ 

id)     We  find  [p//]  =  15.06. 

The  solution  of  the  normal  equations,  with  the  extra  column  for  [p//] 
added,  would  be,  according  to  the  foregoing  scheme, 


X 

y 

- 

12 

0 
+  II 

-  7 

-  4 
+  15 

-  5-32      ' 

-  6.64 
+  11.96 
+  15.06 

'  2.36 

I 

0 
+  II 

-'  0.583 
-  4 
+  10.917 

-  0.443  J 

-  6.64 
+    8.857 
+  12.70 

(=  15.06  -  2.36) 

I 

-  0.364 
+  9-462 

—    0.604 
+    6.422 
+    8.69 

(=  12.70-  4.01) 

[Vt't.]  = 

+    0.68 
+    4-30 

(=    8.69  -  4.39) 

Mean  value  of  [pv-]  =  4.29. 
Hence  (Art.  105) 


-v/M=-- 


and  (see  Ex.,  Art.  103) 


M.  =  1.47"  Vo. 1 193,      A^  =  1.47"  \/o.  1049,      M,  =  1.47"  Vo. 1 05 7 

=  0.51"  =  0.48"  =  0.48" 


INDIRECT    OBSERVATIOXS    OF    ONE    UNKNOWN        137 

Ex.  2.  —  Show  that 

[A7']  =  [v'-]. 
[Multiply  equations  i,  Art.  105,  by .7/1,  v.,,  .  .  .  and  add.     Then,  since, 
[av]  =  o,      [(^7/]  =  o,       .  .  , 
.-.  [At/]  =  -  [Iv]  I 
Ex.  3=  —  Show  that 

[7.=]  =  [A^l  -  I^  - -t^':^' 

l-    J       ^     •'        [aa]        [bb.i] 

[Form  the  normal  equations  from  equations  (3),  Art.  105,  and 

[ad]  (.1-0  -  X)  +  [ab]  ( Jo  -  j)  +  •  •  ■  =  [« -^], 

[a^]  U-o  -  ;t-)  +  [bb]  ( Jo  -J)  +  .  .  .  =  [^  4 


since  [«7/]  =  [(^7']  =  •  •  •  =  o. 

Hence,  from  Art.  106, 


Ex.  4.  —  From  the  equation 

[a/]x+[b/]j+ [//]  =  -[7'=], 


and 


deduce 


[a/]       [b/.i]  „       U-/.2]  „ 
\ad\      \bb.\\  {cc.2] 


[.^]  =  [//]    r^^J^    ^''■''^' 


\aa\        \bb.  i  ] 
Ex.  5.  —  Prove  that       [a?/]  =  [/37/]  =  .  .  .  =  c 
Ex.  6.  — From 

[«A]2  [^A.l]2 


M  =  M- 


deduce  the  formula 


2        [^^-1 
If'  =  -^ — J — 

«  —  Hi 


108.  To  Find  the  Precision  of  any  Function  of  the  Ad- 
justed Values  X,  Y,  .  .  .  —  This  is  the  more  general  ease  of 
the  problem  just  discussed.     The  method  of  solution  is : 

First,  to  find  r,  the  p.  e.  of  an  observation  of  weig^ht  unity, 
and  next//,,  the  weight  of  the  function,  whence  the  p.  e.  of  the 
function  is  given  by 

r  "Slllf,; 

where  u^,  is  the  reciprocal  oi  Pp. 


138  THE    ADJUSTMENT    OP    OBSERVATIONS 

The  value  of  r  is  computed  from  (5)  or  (6),  Art.  105. 
Next,  to  find  tip.      Let  the  function  be 

F=f{X,  F,  .  .  .  )  (i) 

in  which  X,  Y,  .  .  .  are  functions  of  the  independently  observed 
quantities  M^,  M^j  .  .  .  M„. 

Reducing  the  function  to  the  linear  form,  we  have,  adopting 
the  notation  of  Art.  74, 

=/(X\  Y',.  .  .)  +  g^,a+^^  +  ...  (2) 

or,  as  it  may  be  written, 

dF=  G,x+G,y+  ...  (3) 

Now,  since  x,  y,  .  .  .  are  not  independent,  but  are  connected  by 
the  equations 

]aa\x  +  [ab'\y  +  .  .  •  =  [al], 
{ab-]x  +  [M^]y+  .  .  .  =[/'/], 


we  must  get  rid  of  this  entanglement  by  expressing  these  quan- 
tities X,  y,  .  .  .  in  terms  of  l^,  l.y,  .  .  .  ,  which  are  independent 
of  each  other.     From  Arts.  96-97  we  may  write 

X  =  [a/],     y  =  [^/],  .  .  . 

where  a^,  a,„  ...;  ySj,  /3,,,  ...;..  .  are  functions  of  a^,  b^, 
.  ,  .   ;  a.,,  b.„  ...;...      Hence,  substituting  in  (3), 

dF=  {G,a,  +  6^3/3,  +  .  .  .)/^  +  {G^a^_  +  G.^^.^  +  .  .  .)  /^  +  .  .  . 

and,  therefore  (Arts.  13,  47),  since  the  observation  equations  have 
been  reduced  to  weight  unity, 

Up  =  (G,a,  +  G.ft,  +  •••)'  +  (<^i«2  +  <^2y8,  +  •••)'  +  •• 

=  G,'[aa]  +  2  G,G.[a(3]+  •    •    • 

+  G2'     W]  +  •  •  •  (4) 

+  •    •   • 

where  [aa],  [a/3]  .  .  .  may  be  found  in  the  manner  indicated 
in  Arts.  98,  loi,  or  103.     Hence  tfp  is  known. 


INDIRECT    OBSERVATIOXS    OF    ONE    UNKNOWN        139 

109.    {b)  Eq.  4  may  be  written 

uj,  =  G,Q,  +  G,Q,  +  •  •  ■  +  C;„a,  (S) 

where 

<2,  =  [aa]  G,  +  [ay8]   6^,  +  •   •   • 

(2,  =  [a/3]6^,  +  [)8)8](?3  +  .   ..  (6) 


that  is,  where  (see  Eq.  i,  Art.  103) 

G,  =  \_aa]  Q,  +  [ab-]  (2^  +  •  •  • 

G,  =  [ad]  Q,  +  [M]  Q,  +  .  .  .  (7) 


Hence  the  weight  of  a  function 

G,x  +  G,y  +  •  .  . 

of  several  independent  unknowns  x,  y,  .  .  .  is  found  from 

nF  =  [GQl 

where  Q^,  Q.^,  .   .   •  satisfy  the  equations 

[aa]  Q,  +  [a/>]  Q,  +  .  ■  ■  =  G„ 
[ab]Q,  +  [M]Q,+  -  ■  ■  =  G„ 


Therefore,  we  conckide  that,  if  in  a  scries  of  obseii'ation  equations 
the  values  of  the  unknoivns  x,  y,  .  .  .  are  required,  as  well  as 
their  weights  or  the  weight  of  any  function  of  them,  these  results 
can  be  found  at  one  time  by  making  a  solution  of  the  normal 
equations  for  finding  x,  y,  .  .  .  in  general  terms,  and  then  sub- 
stituting for  [al],  [bl],  .  .  .  their  numerical  values  on  the  one 
hand  and  the  values  of  G^,  G.,,   .  .  .  on  the  other. 

1 10.  {c)  This  result  may  be  stated  in  other  forms.  Thus, 
from  Eq.  4,  by  substituting  for  [aa'],  [a/3],  .  .  .  their  values 
from  Art.  loi  or  by  substituting  for  <2,.  (?,.  •  •  •  their  values  in 
(6)  as  expressed  in  Art.  loi  we  have,  after  a  simple  reduction, 

G,^       {G,R,  +  G.;f      {G,R,  +  G,S,+  G,y 
"^  =  [aa]  +  ^    [MTi]  +  [rr:2]  +  ^"^ 

Comparing  this  expression  with  {d).  Art.  106,  it  is  evident  that 


140 


THE    ADJUSTMENT    OF    OBSERVATIONS 


the  several  terms  are  such  as  would  result  from  the  following 
elimination  (Ex.  three  unknowns)  by  finding  the  products  of  the 
quantities  bracketed  : 


X 

y 

z 

Vab] 
[ac] 

[be] 

[ac] 
[be] 
[cc] 

- 

.G, 
[aa] 
'  G,J^,  +  G, 

G^J?,  +  G2 
[bb.i] 
■  G.R^  +  G^S^  +  6^3 

I 

[ab] 
[aa\ 

Vbc.i] 

[ac] 
iaa] 
Vbc.i] 

[CC.l] 

I 

[bc.i] 
[bb.i] 

[CC.2] 

I 

'  G,Ro+  G.S.  +  G^ 

[CC.2] 

III.    {d)    The  expression  (8)  for //y^  may  be  easily  transformed 
into 

k^^  \aa\  +  k^'lbb.i]  +  k:-  \cc.2\  +  •  •  . 
where 

{ab\k^-^  \bh.T.\k,  =-  G., 

[ac]  ^Q  +  [bc.i]  k^  +  ^^.2]  k.,  =  —  6^3, 

Circumstances  must  decide  which  of  the  four  forms  given  should 
be  chosen  in  any  special  case.  A  machine  can  be  used  to  the 
best  advantage  with  the  second  and  third  forms.  The  third 
form  is  also  convenient  when  the  weights  alone  are  required, 
without  the  values  of  the  unknowns,  and  the  second  when  the 
values  of  the  unknowns  can  be  found  by  an  easier  method  of 
solution  than  the  Gaussian  method  of  substitution. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        141 

112.   Ex.  I.  —  In  Ex.  4,  Art.  77,  it  is  required  to  find  the  m.  s.  e.  of  the 
angle  PSB. 

The  function  is 

dF  =  -  x  +  z. 

First  Solution.     From  equation  (4), 
up  =  [aa]  -  2  \p.y]  +  [77] 

=  0.1 193  —  2  X  0.0616  +  0.1057  (from  Ex.,  An.  103) 
=  0.1018 
Second  Solution.     From  equations  (7), 

+  122.  -    7  2,1=  -I 

-    7  2i-    4^2  +  15^3=  +  I 
Hence 

Qi=  -  0.0577,    03  =  +  0.0440 
and 

up  =  —   I  X  —  0.0577  +  I  X  0.0440 
=  0.1017 

Third  Solution.     Add  the  extra  column  G  to  the  solution  of  the  normal 
equations,  which  would  give  the  scheme 


X 

y 

z 

G 

+  12 

+  II 

-  7 

-  4 
+  15 

—  I 

0 

+  I 

—  0.0833 

0     : 
+  0.4169 

0 

+  0.4169  ■ 

+  0.0441 

■ 

+   I 

+  II 

-  0-5833 

-4- 

+  10.9169 

+  I 

-  0.3636 
+  9-4625 

+  I 

Hence  «/-•  =  —  i  X  —  0.0S33  +  0.4169  x  0.0441 

=  0.1017 
Fourth  Solution. 

12  /&„  =  +  I 

II  ^,  =0 

-  y  k^-    4  /{',  +  9 4625  k^=  -  \ 


142  THE    ADJUSTMENT    OF    OBSERVATIONS 

.'.  k^  =  0.0833,  /tj  =  o,  /&2  =  -  0.0440 

UF=  (0.0833)2  X   12  +  (0.0440)'  X  9.4625 

=  O.IOI6 

Also  />iy=  i.47"\/o.io2 

=  0.47". 

Ex.  2.  —  Given  the  observation  equations 

a^x  +  b^y  =  /j, 
UnX  +  b.^y  =  I21 


Un-v  +  b„y  =  4, 
to  find  the  v^eight  of  fx  +  gy- 
[The  normal  equations  are 

[aa\x  +  [ab]y  =  [al], 
iab]x+{bb}y  =  [bl]. 

■  -^  =  wiw^x  ^^''^^'--  ^""'^  f^^^  +  ^'''^''^■'^ 

1 13.  To  find  the  average  value  of  the  ratio  of  the  weight  of  the 
observed  value  of  a  quantity  to  that  of  its  adjusted  value  in  a 
system  of  independently  observed  quantities. 

The  adjusted  value  of  the  first  observed  quantity  M^  is  M^  + 
V .  From  Art.  74  it  follows  that  the  weight  of  M^  +  v^  is  the 
same  as  the  weight  of  l^  +  Vy     Now, 

/j  +  z'l  =  a^x  +  b^y  ■\-  ■  •  •  (i) 

Hence  if  P^  is  the  weight  of  the  adjusted  value  M^  +  v^  that  is, 
is  the  weight  of  the  function  a^x  +  b^y  +  .  .  .,  and/^, /^  •  •  • 
are  the  weights  of  l^  Z^,  .  .  ,  ,  we  have 

~  =  a,Q,  +  b,Q,^-.  ■  .  (2) 

-Ml 

where  (see  Eq.  4,  Arts.  104,  109) 

Q^  =  \uaa\  a^  +  \tlo.^\  b^-\-  •  ■  ■  =  U^a.^, 

Q^  =  [uap]  a,  +  {^m  l\  +  -  ■  ■=  'h^v  (3) 


Therefore  by  substitution  of  Q^  Q.,   -  •  ■  in  (2), 


•VI 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN       143 
Similarly 


Hence  by  addition 

r/|  =  [,/a]  +  [I'li]  +  .  .  .  to  ;/i  terms 

=  ;/,-,  the  number  of  independent  unknowns. 
Hence  the  average  value  of  \_p/P']  is  njn,  or  in  words,  the 
average  value  of  the  ratio  of  the  zveight  of  the  observed  value  of 
the  quantity  to  that  of  its  adjusted  value  is  the  ratio  of  the 
number  of  independent  unknoions  to  the  number  of  observed 
quantities. 

This  result  may  be   very  readily  derived  directly  as  follows : 
In  (I)  put  [a/]  for  x,  [/3/]  for  j,   .  .   .  ,  and  we  have 

/,  +  V,  =  {a,a,  +  b,P,  +  ■■   ■)l,  +  {'ha,  +  l>,(i,  +  ...)!,+  ... 

Hence,  since  l^,  /„,  ...  are  independent, 


I  ,  .         .      „         .  vo     I 


-^  =  {a, a,  +  ^/3,  +  •  •  0^^  +  i'h'H  +  1>A  +  •  •  •  fj^  + 

=  rtj  f  [«aa]  a^  +  [uafi]  l\  +  ■   ■   ■   \ 

+  b,\{uaP]a,  +  [HmK  +  -  ■  ■  \ 


=  u,  (a^a,  +  b,(3,  +  •  •  • )  (4) 

as  before. 

Ex.  — To  check  the  weights  of  the  adjusted  vaUies  of  the  angles  found 
in  Ex.  4,  Art.  77. 

The  weights  of  the  measured  values  of  the  angles  are 

5,         7,         4,         7,         4 
The  weights  of  the  adjusted  values  are  (Ex.,  Art.  100  and  Art.  112), 
8.4,        9-5.        9-5.        9-8,        7-5 

^•-  8-4  +  ^ + ^5  +  P  +  h 

=  3 

=  the  numljcr  of  independent  unknowns, 

as  it  should. 


144  THE    ADJUSTMENT    OF    OBSERVATIONS 

Tivo  Special  Artifices. 
114.  The  labor  of  solving  and  finding  the  values  of  the  un- 
knowns may  be  often  shortened  by  taking  advantage  of  some 
principle  inherent  in  the  form  of  the  observation  equations  them- 
selves. For  example,  if  we  have  a  series  of  observation  equations 
containing  two  unknowns,  and  of  which  the  coefficient  of  the 
first  unknown  is  unity,  instead  of  solving  in  the  usual  way  we 
may  reduce  the  observation  equations  to  equations  containing 
the  second  unknown  only,  and  thus  solve  more  readily. 
Given 

x  +  b^y  =  ly,  weight /i, 
x  +  b^y  =  4  weight  A. 

Forming  the  normal  equations  in  the  usual  way,  we  have 

[/]    .Y  +  [//;]  J  =  [//], 
Ipb-X  X  +  \pF^\y  =  [pb/], 

whence  eliminating  x, 

y  \[pi^Y  -  [/] [pf>']\  =  [/^l [/^']  -  [/]  t^^''']- 
Now,  if  the  first  normal  equation  is  divided  by  [/],  so  that 

^  [/]  -^    [/] 

and  from  this  equation  each  of  the  observation  equations  in  suc- 
cession is  subtracted,  there  result  the  equations. 

The  normal  equation  for  finding  j  from  these  equations  is, 

\{p]  [#']  -  [#']|  y  =  IP\  \P^'^\  -  [^^]  t^^l' 

the  same  as  results    from  the  elimination  of  x  in  the  normal 
equations. 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        145 

This  process  is  specially  convenient  if  the  original  observation 
equations  are  numerous,  and  the  coefficients  /;, ,  b.,,  .  .  .  and  the 
terms  /,,/,,   .   .   .  are  large  and  not  widely  different. 

Ex.  —  To  solve  the  equations  in  Ex.  3,  Art.  77. 
The  mean  of  the  equations  is 

x  -  o.C)jy  =  -  1.97. 
Subtract  each  equation  from  this  mean  equation,  and 

+  0.62  y  =+  o.oi, 
+  0.47  J  =-  0-07, 
4-  0.26  y  =  +  0.03, 

—  0.01  y  =  —  0.02, 

—  0.25  _y  =  +  0.03, 

—  1.08  J/  =      0.00. 

The  normal  equation  formed  from  these  equations  is 

+  I  qiy  =  —  0.27, 
and  .'.  y  =—  0.014^ 

Substitute  for/  its  value  in  the  mean  equation,  and 

X  =  -  1 .95'. 

115.  Again,  we  may  take  advantage  of  the  presence  in  the 
problem  of  some  arbitrary  quantity  to  which  a  convenient  value 
may  be  assigned.  Thus,  to  find  the  difference  of  the  coeffi- 
cients of  expansion  of  two  standards  A  and  B  from  observed 
differences  of  length  at  certain  fixed  temperatures. 

Let      X  =  the  excess  in  length  of  A  over  B  at  an  arbitrary  tem- 
perature /„, 
y  =  the  excess  of  the  coefficient  of  expansion  of  A  over  that 
of  ^, 
1^,1^,.  .  .  =  the  observed  differences  in  length  at  temperatures  /,,  /.,, 
.  .  .  and  whose  weights  are/j,/.^,  .  .  . 

We  have  then  the  observation  equations 

x  +  (f.-  Qy  -  /,  =  z/j,  weight/,, 

X  +  (4  -  Qy  -  /,  =  7'.,,  weight  /.,,  (i) 


146  THE    ADJUSTMENT    OF    OBSERVATIONS 

and  the  normal  equations 

[p]x  +  {[pn-[j>]Qy  =  [j>n 
\[p]  -  [/J  ^0^  ^^-  +  L/  (^  -  ^oT-]!   =  [(^  -  o pn-        (2) 

As  the  vakie  of  /„  is  arbitrary,  the  normal  equations  will  be 
simplified  by  taking  it  equal  to  the  weighted  mean  of  the  tem- 
peratures ;  that  is, 

/  -  ^  •  (3) 

and  they  will  then  become 

[P]x  =  {pl\ 
\P{t-t,y]y  =  {{f-Qpll  (4) 

from  which  x  and  y  are  found  at  once,  with  their  weights  at  the 
mean  temperature  t^. 

If  the  values  of  /  are  numerically  large,  it  will  lessen  the  labor 
of  finding  the  value  of  j  if  the  mean  value  of  x  found  from 

is  substituted  in  the  observation  equations  before  the  normal 
equation  in  y  is  formed.     We  should  then  have 

{p{i-  t,y\y  =  {p{f-  Q(J -  ^)l 

from  which  to  find  y. 

It  is  evident  that  the  value  of  j  found  in  this  way  is  the  same 
as  before.     For 

[/  {f  -  Q  (^  -  ^^Ol  =  {P(i-  O  ^]  -  \lPf^  -  [^1  ^ol  ^ 

since  the  coefficient  of  x  is  equal  to  zero  from  Eq.  3. 

The  quantity  /  —  x  comes  in  very  conveniently  in  computing 
the  residuals  v  in  finding  the  precision. 

The  Precision. If   Ji    is  the    number  of    observations,   the 

number  of  unknowns  being  2,  we  have  for  the  m.  s.  e.  /i  of  a 
single  observation, 


INDIRECT    OBSERVATIONS    OF    ONE    UNKNOWN        147 


/  [/>vz>] 


and 


[/] 


/*y  = 


The  length  at  any  temperature  /'  is 

x+  {f'  -  O  }', 
and  its  m.  s.  e.  /i^  is  found  from 

_  ^  ,     (^'  —  O^     2 
The  weight  is  greatest  when  ft^  is  least,  that  is,  when 

°    [/I 

116.  Ex.— The  following  were  among  the  observations  made  for  the 
determination  of  the  difference  of  length  between  the  Lake  Survey  Standard 
Bar  and  Yard  ;  and  also  for  the  difference  between  their  coefficients  of  ex- 
pansion.    The  unit  is  tooW?  i^^ch. 

Required  the  difference  of  length  at  62'  Fahr.  and  at  any  other  tempera- 
ture /"• 


D.MK. 

Observed  Temp.  (/). 

Bar  —  Yard  (/). 

Weight  (/) 

1872,  March    5      .     .     . 

0 
24.7 

791 

5 

"       14      .     .     . 

37-' 

811 

1 

"       26      .     .     . 

61.7 

833 

6 

April     4      .     .     . 

49-3 

S20 

6 
8 

"       12      ... 

66.8 

847 

"       20      .     .     • 

71-5 

S.)., 

S 

Let  X  =  the  most  probable  difference  between  Bar  and  Yard  at  62^  Fahr. 
y  =  the  most  probable  difference  between  coefficients  of  expansion 
of  Bar  and  Yard. 
The  observation  equations  will  be  of  the  form 

X  +  {/  -  62)/  -  I  =  v. 


148  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  computation  is  arranged  in  tabular  form  as  follows  : 


p 

pt 

Pl 

/-/« 

I-  X 

5 

1-3-5 

3955 

-32.2 

-40 

I 

37-1 

811 

—  19.8 

—  20 

6 

370.2 

4998 

+    4-8 

+    2 

6 

295.8 

4920 

-     7.6 

—  II 

8 

534-4 

6776 

+    9-9 

+  16 

8 
34 

572.0 

6792 
28,252 

+  14-6 

+  18 

1933-0 

4  =  56.9° 

A'  =  831 

p{t-  tj 

P(t  -  4)  (/  -  -r) 

y  {t  -  A,) 

7' 

pvv 

5184.20 

6440.0 

—  40.6 

-0.6 

1.8 

392.04 

396.0 

-  24.9 

-4.9 

24.0 

138.24 

57.6 

•4-  6.0 

+  4.0 

96.0 

346.56 

501.6 

-  9.6 

+    1.4 

11.8 

784.08 

1267.2 

+  12.5 

-3-5 

98.0 

1705.28 

2102.4 

+  18.4 

+  0.4 

1-3 

8550.40 

10,764.8 

232.9 

y  = 

1.26 

M«  = 


ih 


1-3, 


Value  of 


V6-2 
7-6^ 
V34 

7-6 
\/855o.40 
X  =  831         at 
6.4 


7.6, 


56.9° 

5-1° 
62.0° 


=  837.4 

^63^=    (1.3)'+    (5-1)'    X(0.I)==    1.9. 

These  values  may  be  checked  by  computing  by  the  ordinary  process. 


CHAPTER   V 

ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS 

117.  We  now  take  up  the  third  division  of  the  subject  as 
laid  down  in  Art.  25.  So  far  the  quantities  we  have  dealt  with, 
whether  directly  observed  or  functions  of  the  quantities  ob- 
served, have  been  independent  of  one  another ;  but  if  they  are 
not  independent  of  one  another  —  that  is,  if  they  must  satisfy 
exactly  certain  relations  that  exist  a  priori  and  are  entirely 
separate  from  any  relations  demanded  by  observation  —  they 
are  said  to  be  conditioned  by  these  relations,  or  the  relations 
are  spoken  of  as  conditions. 

All  problems  relating  to  conditioned  observations  may  be 
solved  by  the  rules  laid  down  in  the  preceding  chapters. 

Let,  with  the  usual  notation,  X^,  X^,  .  .  .  X^  denote  the 
most  probable  values  of  n  directly  observed  quantities  M^,  Hf.,, 
.  .  .  Mj^  whose  weights  are  p^,  p.,,  .  .  .  /„  respectively.  Let 
the  «p  conditions  to  be  satisfied  exactly  by  the  most  probable 
values,  when  expressed  by  equations  reduced  to  the  linear 
form,  be 

a'X^  +  a"X.,  -j-  .  •  •  -  Z'  =  o, 

b'X^  +  b"X.,  +  •  •  .-  L"  =  0,  (i) 


where  a',  a",  .  .  .  ',  b',  b" ,  ...;...;  //,  L",  .  .  .  arc  known 
constants. 

If  2\,  V.,,  ...■?'„  denote  the  most  probable  corrections  to  the 
observed  values,  so  that 

X,-M,  =  V,, 
(2) 

Xn  -  M„  =  V„, 
149 


150  THE    ADJUSTMENT    OP    OBSERVATIONS 

we  have  the  reduced  condition  equations 

a'v^  +  «"z'2  +  •  •  •  -  /'  =0, 
b'v^  +  b"v''  +...-/"  =  o, 

or  \av\  —  I'  —  o, 

[bv]  -  I"  =  o,  (3) 


where  V  ^V  -  [aM],  I"  =  L"  -  \bM\  .  .  .  ,  and  are,  there- 
fore, known  quantities. 

The  most  probable  system  of  corrections  is  that  which  makes 

[^z;2j  _  3^  minimum,  w  suppose. 

The  problem  is  to  solve  this  minimum  function  when  the 
corrections  v  are  subject  to  the  above  n^  conditions. 

Direct  Solutioji  —  Method  of  Independent  Unk7iowns. 

118.  It  is  plain  that  n^  of  the  corrections  can,  by  means  of 
the  condition  equations,  be  expressed  in  terms  of  the  remain- 
ing n  —  n^  corrections,  and  that  by  substituting  these  n^  values 
in  the  minimum  function,  we  should  have  a  reduced  minimum 
function  containing  n  —  n^  independent  unknowns.  This  func- 
tion can  be  found  in  the  usual  way  by  equating  to  zero  its 
differential  coefficients  with  respect  to  each  unknown  in  suc- 
cession. The  «  —  n^  resulting  equations,  taken  in  connection 
with  the  71^  condition  equations,  determine  the  n  corrections  v^, 
v^,  .  .  .  %r^.     Thence  \^pzn'']  is  found. 

The  solution  of  the  n  —  n^  equations  can  be  carried  through 
by  any  of  the  methods  of  Chapter  IV.  The  precision  of  the 
adjusted  values,  or  of  any  function  of  them,  can  also  be  found 
as  in  Chapter  IV. 

Ex.  I.  —  Take  that  already  solved  in  Ex.  4,  Art.  77. 

Let  v^,  7>o,  Vs,  7/4,  7/5  be  the  most  probable  corrections  to  the  measured 
angles,  then  the  conditions  to  be  satisfied  are 

FSB  +v,  =  FSB  +  7/3  -  FSF  -  7/, , 
OSB  +  v.  =  FSB  +  7/3  -  FSO  -  v^. 


ADJUSTMENT    OF    CONDITIOXED    OBSERVATION'S       151 

Substituting  for  PSB^FSB,  etc.,  their  measured  values,  the  condition  equa- 
tions may  be  written 

Vi-  v.^  +  7/4  =  -  0.76, 

V2  —  v^+  v^=^  —  1 .66, 
with  5  7'f  +  7  v^  +  A'vi  +  1  Vi  +  4  •z/j'  =  a  min. 

Substitute  for  7/^,  v-^  in  the  minimum  equation,  and 

5  ff  +  7  v^-  +  4  ■^s^  +  7  {Vi  —  "^3  +  0.76)2  +  4  (7/0  -  •6/3  +  1.66)^  =  a  min. 

Hence,  differentiating  with  respect  to  t/,,  v.,,  7/3,  as  independent  variables, 
we  have  the  normal  equations 

\2Vi  -    iv_,  =  -  5.32, 

1 1  7/2  -    47/3  =  -  6.64, 
-  7  7'i  -  47/,  +  15  7/3  =     11.96, 
whence  7\  =  —  0.05",       '^'2  =  ~  o-36",       ^'3  =  +  0.68"; 

and  from  the  condition  equations, 

7/,  =  -  0.03",        7/5  =  -  0.62". 

These  results  are  the  same  as  those  already  found  in  Art.  77. 

Ex.  2.  —  The  angles  A,  B,  C,  of  a  spherical  triangle  are  equally  well 
measured  ;  required  the  adjusted  values  and  their  weights. 

The  condition  equation  to  be  satisfied  is 

A  +  B  +  C=  iSo  +  e,  (i) 

where  e  is  the  spherical  excess  of  the  triangle. 

Putting  Ml  +  7/1,  M2  +  7/2,  M3  +  z/gfor  A,  B,  C,  the  condition  equation 
becomes 

7/1  +  7/2  +  7/3  =  180  +  e  —  [M] 

=  /,  suppose.  (2) 

Also  7',^  +  7/2^  +  7/3^  =  a  min. 

Substitute  for  7/3  from  (2)  in  the  minimum  function,  and 

7^1'  +  -!'■!  +  (■?^i  +7/2  -  /)^  =  a  min. 

Differentiating  with  respect  to  the  independent  variables  7/,,  v.,  and 

(3) 

/ 

which  give 

Also,  from  Eq.  2, 

Hence  f/ie  co7'rection  to  each  ani^le  is  one-third  of  the  dijjference  of  the  theo- 
retical and  meastcred  suvis  of  the  three  an  tries. 


^1 

+ 

7/0  = 

=  /, 

^1 

+ 

27/2  = 

=  /, 

'V^ 

1  ~ 

7/2  = 

I  ^ 
3' 

7/j 

/ 

'~3' 

152  THE    ADJUSTMENT    OP    OBSERVATIONS 

To  find  the  weight  of  the  adjusted  value  of  an  angle,  as  A. 
The  function  is  dF  =  7'i . 

Hence,  following  the  method  of  Art.  loi  {b), 

where  G^  =  i,  and  Q^,  Q.,  are  found  from 

2  2.  +      Q.  =  I, 

that  is,  weight  of  ^  =  |  if  weight  of  measured  value  is  unity. 

Check.     Weight  of  direct  measure  oi  A =  i, 

Wt.  of  indirect  meas.  (=  i8o  +  e  —  ^  —  C)  of  ^  =  i. 

Weight  of  mean =  li. 

as  already  found. 

Ex.  3.  —  To  find  the  weight  of  a  side,  «,  in  a  triangle,  all  of  whose  angles 
have  been  equally  well  measured,  the  base,  b,  being  free  from  error. 

rr  ,sin^ 

Here  F  =  a  =  b  -. — 5 , 

sm  B 

:.  dF  =  a  sin  \"  (cot  Av-^  —  cot  Bv^. 
The  weight  is  found  from 

Up  =  a  sin  \"  cot  AQi  —  a  sin  i"  cot  BQ^, 

where  Q^,  ^2  satisfy  the  equations  (Art.  loi), 

-  Qi  +     Q2  =      ^  sin  i"  cot  A, 
Qi  +  2  Q,  =  —  a  sin  i"  cot  B. 

Hence,  Wjj,  =  §  «=*  sin^  i"  (cot^  A  +  cot^  B  +  cot  A  cot  B). 

Indirect  Solution  —  MetJiod  of  Correlates. 

119.  If  the  unknowns  in  the  condition  equations  are  much 
entangled  the  direct  sohition  would  be  very  laborious.  It  is  in 
general,  therefore,  advisable,  instead  of  eliminating  the  n^  un- 
knowns directly,  to  do  so  indirectly  by  means  of  undetermined 
multipliers,  or  correlates,  as  they  are  called. 

If  we  multiply  the  condition  equations  (3),  Art.  117,  in  order 
by  the  correlates  C^,  C,,  .  .  . ,  we  may  write 

o)  =  C,  ([«7']  -  /')  +  C,  {^bTi\  -  I")  +  .  •  •  -h  \pvv\  =  a  min.      (i) 

and  determine  C^,  C,  .  .  . ,  accordingly. 
By  differentiation, 


ADJUSTMENT    OP    CONDITIONED    OBSERVATIONS 


'53 


i?) 


If  we  place  equal  to  zero  the  coefficients  of  n^  of  the  differen- 
tials di\,  dv.,,  .  .  .,  we  shall  have  ;Zp  equations  from  which  to 
find  C^,  C,,  .  .  .  Substitute  these  7i^.  values  in  the  expression 
for  dco,  and  there  will  remain  ;/  —  n^  differentials  which  are 
independent  of  one  another.  In  order  that  the  function  may 
satisfy  the  condition  of  a  minimum,  the  coefficients  of  each  of 
these  differentials  must  be  equal  to  zero.  This  gives  u  —  ;/,. 
equations,  which  equations,  taken  in  connection  with  the  //,.  con- 
dition equations,  give  the  n  unknowns  z\,  v.„   .  .  .  t'„. 

The  practical  solution  would,  therefore,  be  :  Form  n  equations 
by  placing  equal  to  zero  the  differential  coefficients  of  the  mini- 
mum function  with  respect  to  each  of  the  quantities  v^,  v.,,  .  .  . 
v^.  From  these  n  equations  and  the  n^  condition  equations 
determine  the  «  +  71^  unknowns  C^,  C,  .  .  .  ,  i\,  v.,,  .  .  .,  and 
thence  the  function  \^pvv]. 

In  carrying  this  out  the  form  of  the  differential  equation  (2) 
shows  that  it  would  be  advantageous  to  multiply  the  minimum 
equation  by  —  \,  and  so  write  (i)  in  the  form 

q  {[av]  -  /')  +  q  ([^^'1  -  /")  +  ■  ■     -h  [PM  =  a  niin.        (s) 
Differentiating,  we  have  the  n  correlate  equations 


a"  C\  + />"€..  +  .  .  •=/,7'2, 


(4) 


Substituting  for  r^,  v.^,  ...  in  the  condition  equations  their 
values  derived  from  these  equations,  and  the  normal  c(|uati(>ns 
result.     They  are 


~aa 

ra/n 

<'l 

+ 

lp\ 

L/J 

'a/>~\ 

'/>/>! 

(\ 

4- 

L/J 

./J 

c,  +  . .  .  =  /', 
c:.  +  .  .  .  =  /", 


Cs) 


154 


THE    ADJUSTMENT    OF    OBSERVATIONS 


Solving,  we  obtain  C^,  Co,  •  •  • ,  and  thence  v^,  v^, 
and  X^,  X^,  .  .   .  from  (2),  Art.  117. 
The  normal  equations  may  be  written 

\iiad\  Cj  +  \iiab\  C^-\-  •  ■  ■  =  I', 
[uab]  C;  +  [ubb]  C  +  -  •  •  =  /", 


from  (4), 


(6) 


where  ?/j,  ?/„»  •  •  •  denote  the  reciprocals  of  the  weights  /p />.,, 
.  .  .  The  form  of  these  equations  shows  that  the  coefificients 
[tiaa],  \tiab\  .  .  .  may  be  computed  as  in  Art.  79,  the  corre- 
sponding scheme  being, 


c, 

a 

Q 

.    .    . 

"^'x 

»l 

a' 

b' 

c' 

7/2 

«2 

a" 

b" 

c" 

If  the  elimination  of  the  normal  equations  is  performed  by 
the  method  of  substitution  (Art.  84),  we  have,  by  collecting  the 
first  equations  of  the  successive  groups, 

[uaa]  Ci  +  [uab]     C,  +  [uac]     C,  +  ■  •  •  =  /' 

+  [ubb.i]  C,  +  [Jibc.i]  C\  +  .  .  .  =  I".x 
'  +  \UCC.2\  C3  +  •  •  .  =  I'". 2 


(7) 


where   V ,  I" .\,   I'" .2,    correspond    to    \al\   \bl .\\   \cl .2\   .  .  . 
respectively. 

These  equations  being  precisely  similar   in   form   to   Eq.  8, 
Art.  84,  the  elimination  gives  (see  Art.  loi), 


C,= 


a  = 


I"  I  /'" 


\ubb.\\ 
\ubb.\\ 


.R"  + 


\lUC.2\ 

I'"  -> 

\llCC.2\ 


(8) 


where 


[ua(7\ 


ADJUSTMENT   OF    CONDITIONED    OBSERVATIONS      155 

o  =  [^  +  [^,^'  +  ^".  (9) 

[uaa]       [u/'P.i] 


r.i  =  I'R'  +  / ', 

/'".2  =  I'R"  +  l"S"  +  /'",  (10) 


120.    Ex.  I.  —  Take  that  solved  in  Ex.  i,  Art.  118. 
The  condition  equations  are 

7/1  -  7/3  +  7^4  =  -  0.76, 

7/3  —  7/3  +  7/5  =  —  1.66.  (i) 

The  correlate  equations  consequently  are 

Ci  =  5  ^1 . 

-  Q  -  C2  =  4^3.  (2) 

Q    =    7^4, 

a  =  47/5. 

To  form  the  normal  equations,  we  may  substitute  for  7/,,  ta,  .  .  .  from  (2) 
in  (i),  or  proceed  by  means  of  the  tabular  form  in  Art.  1 19.     We  find 
0.59  Ci  +  0.25  C  =  -  0.76, 
0.25  C"i  +  0.64  C  =  —  1.66. 
The  solution  of  these  equations  gives 

C,  =  -  0.23,  Q  =  -  2.49. 

whence,  from  the  correlate  equations, 

7/,  =  -  0.05",      7/2  =  -  0.36",      7^3  =  +  0.68",      7/4  =  -  0.03",      7/5  =  -  0.62". 
Check.  —  The  results  satisfy  the  condition  equations. 
Ex.  2.  — The  angles  A,  B,  C,  of  a  spherical  triangle  are  measured  with 
their  weights,  j?^,, /_,, /.,;  required  their  adjusted  values. 

The  condition  equation  may  be  written  (see  Ex.  2,  Art.  118) 

7/1   +  T'g  +  ^At  =  A 

with  [pv"^  =  a  min. 

The  correlate  equations  are  C  =  ^,7/, , 

C  =  P.V2 , 

C  =  p^v^ , 
and  the  normal  equation  [u]  C  --=  /. 


/,       7/.,  =  ,    ,  /,       v..  =  ,    ,  /. 


Hence  the  adjusted  values  are  known. 
When  the  weights  are  equal,  then 


156  THE    ADJUSTMENT    OF    OBSERVATIONS 

the  same  results  as  in  Ex.  2,  Art.  118. 

Note.  —  If  a  condition  equation  is  of  the  form 

«,7/l    +  a.fO.,  +    •    •    •    +   ttnVn  =   /, 

the  weights  of  the  measured  values  being  p^,p., .  .  .  pn,  then,  proceeding  as 
in  the  above,  we  have 

,  h  '^'■1  —  7 T   l-i    •    •    • 


'       \iiaa\   '  '       iuaa\ 

This  result  is  very  important,  and  will  be  often  referred  to. 

Ex.  3.  — At  U.S.  Coast  Survey  station,  Pine  Mountain,  the  following  were 
the  angles  observed  between  the  surrounding  stations  in  order  of  azimuth : 
Jocelyne-Deepwater    .     65°     11'  '  52.500",     weights, 
Deepwater-Deakyne  .     66'^     24'     15.553",         "       3» 
Deakyne-Burden    .     .     87°    02'    24.703",         "       3, 
Burden-Jocelyne     .     ,  141^     21'     21.757",         "       i, 

required  their  most  probable  values. 

The  condition  to  be  satisfied  is  that  the  sum  of  the  angles  should  be  360=". 

Now,  ,      ,  o        /  // 

Sum  of  measured  values  =  359°    59     54-5 13 

Theoretical  sum     .     .     .  =  360°    00^    00.000" 

.-.  Residual  error  .     .     .  =  5487" 

Hence,  as  in  the  preceding  example, 

1 

Correction  to  each  of  first  three  angles  =  1  ^_  ,  ^  1  ^  ^  X  5487" 

=  0.914". 
Correction  to  fourth  angle     .     .     .     .     =  1,1,1,-  X  5487 

=  2.744"- 

Ex.  4.  — "In  order  to  find  the  content  of  a  piece  of  ground,  I  measured 
with  a  common  circumferentor  and  chain  the  bearings  and  lengths  of  its 
several  sides.  But  upon  casting  up  the  difference  of  latitude  and  departure, 
I  discovered  that  some  error  had  been  contracted  in  taking  the  dimensions. 
Now,  it  is  required  to  compute  the  area  of  this  inclosure  on  the  most  probable 
supposition  of  this  error. 

"  Let  ABCDE  be  a  survey  accurately  protracted  according  to  the  meas- 
ured lengths  and  bearings  of  the  sides  AB,  EC,  .  .  .  A  the  place  of  begin- 
ning, E  of  ending,  AG  ?i  meridian,  AF,  FE  the  errors  in  latitude  and 
departure.  Now,  the  problem  requires  us  to  make  such  changes  in  the 
positions  of  the  points  B.C,  .  .  .  that  we  may  remove  the  error?,  AF,  FE — 
in  other  words,  that  E  may  coincide  with  A  ;  and  these  changes  must  be 
made  in  the  most  probable  manner.  We  have,  therefore,  to  fulfill  the  three 
following  conditions: 

"All  the  changes  in  departure  must  remove  the  error  in  departure  EF. 

"All  the  changes  in  latitude  must  remove  the  error  in  latitude  AF. 

"The  probability  of  these  changes  must  be  a  maximum." 


ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS       157 

Let  a,,  <?,)  «3,  .  .  •  ;  ^11  ^2)  ^3>  •  •  •  ;  denote  the  measured  lengths  and 
bearings  of  the  sides  AB,  BC,  .  .  .  ,  and  x\,  x\,  x^,  .  .  .  ;  ji,  jo,  y^,  .  .  . 
their  most  probable  corrections. 

Now,  since  the  corrected  latitudes  must  balance,  and  the  corrected  de- 
partures must  also  balance,  we  have  the  conditions 

{Uy  +  Xy)  cos  (d,  +  J,)  +  (^2  +  x^)  cos  (fl,  +  J/,)  +  •  •  •  =  o, 
[a,  +  X,)  sin  («,  +  j/,)  +  {a.  +  :r,)  sin  («2  +J2)  +  •  •  •  =  o, 

or,  reducing  to  the  linear  form. 


cos  e,Xi  -  a,  sin  e,y^  +  cos  d.^:.  -  a.  sin  ^,j',  +  •  •  •  +  [a  cos  tf]  =  o, 
sin  ^,.r,  +  iZi  cos  ^,  j,  +  sin  Soj:,  +  a.,  cos  e.j,  +  •  •  •  +  [a  sin  tf]  =  o, 
with 

[A^"']  +  [^J']  =  3.  minimum, 

the  weights  of  ;i-„  jt^,  .  .  .  ;  j'l,  Jo,  .  •  •  be- 
ing A>  A.  •  ■  •  ;  ?!'  ^2,  .  •  •  respectively. 
Hence  the  correlate  equations, 

+  cos^iC,  +  sin  ^iQ  =  P\^  X\t     ,s 

—  flj  sin  ^,Ci  +  <ZiCOS0iC2  =  ^1,  j'j, 

and  the  normal  equations,  '*'•  ■** 

f  fcos^  ^1      f  g^  sin=  g"!  ]  fPsin  g  cos  d\  _  1"^^  sin  Q  cos  ffl 

iL"^J''L  ^  Jj  '   iL    ^    J  L     ^     J 

I  ["sin  g  cos  e\  _  p-  sin  e  cos  g"]  |  ^  ^  j  fs'n'  ^"1  ^  p'  cos^  g"|  |  ^ 


(U 


=  —  [fl  cos  6\ 


Now  if  we  assume 


=  —  [a  sin  <?]. 


^'  =  i:'  ^=  =  i 


as  "  this  seems  best  to  agree  with  the  imperfections  of  the  common  instru- 
ments used  in  surveying,"  the  normal  equations  reduce  to 

Cy  [a]        =  -  [a  cos  e\ 
C2  [a]  =  -  [a  sin  e\ 
from  which  (7„  C  are  known. 

The  weights  indicated  correspond  to  the  assumption  that  the  errors  of 
measurement  of  the  lengths  aredirecdy,  and  of  the  same  l)earings  inversely, 
proportional  to  the  square  root  of  the  lengths  of  the  corre.sponding  courses. 
The  corrections  t„  x„  .  .  .  ;  J'„  ^2  •  •  -i  are  known  from  (2). 
The  errors  in  latitude  (see  Eq.  1)  now  reduce  to 

\a  cos  e] 
cos  ^,  j'l  -  fl,  sm  «,  J,  =  -  «,  — r^j — ' 


158  THE    ADJUSTMENT    OF    OBSERVATIONS 

[a  cos  d] 
cos  62  X2  —  «2  sin  ^2^2  =  -~  ^2  — r^j — ' 

and  the  errors  in  departure  to 

[a  sin  e] 
sin  ^1  x^  +  <z,  cos  ^1  Ji  =  —  «i  — F-^j —  > 

[a  sin  ^1 
sin  e.;,  X2  +  «2  sin  ^0^2  =  —  <^2  — f-i — ' 


Hence  Bowditch's  rule  for  balancing  a  survey  :  "  Say  as  the  sum  of  all  the 
distances  is  to  each  particular  distance,  so  is  the  whole  error  in  departure 
to  the  correction  of  the  corresponding  departure,  each  correction  being  so 
applied  as  to  diminish  the  whole  error  in  departure.  Proceed  in  same  way 
for  the  correction  in  latitude." 

This  problem  was  proposed  as  a  prize  question  by  Robert  Patterson,  of 
Philadelphia,  in  vol.  i.  No.  3  of  the  Analyst  or  Mathematical  Museum, 
edited  by  Dr.  Adrain,  of  Reading,  Pa.,  and  published  in  1807.  In  vol.  i. 
No.  4  are  two  solutions  —  one  by  Bowditch,  to  whom  the  prize  was  awarded, 
and  the  other  by  Dr.  Adrain.  Adrain's  mode  of  solution  is  nearly  the  same 
as  by  the  ordinary  Gaussian  method.  He  employs  undetermined  multipliers 
or  correlates,  exactly  as  Gauss  subsequently  did.  To  Adrain,  therefore,  is 
due  not  only  the  first  derivation  of  the  exponential  law  of  error,  but  its  first 
application  to  geodetic  work. 


To  Find  the  Precision  of  the  Adjjisted  Values,  or  of  any  Func- 
tion of  them. 

121.    The  method  of  proceeding  is  the  same  as  in  Art.  108. 

The  first  step  is  to  find  r,  the  p.  e.  of  a  single  observation, 
and  next  the  weight, /;?,  of  the  fimction,  whence  the  p.  e.  of  the 
function  is  given  by 

r  V«  F, 

71  f>  being  the  reciprocal  of  the  weight. 

(a)  To  find  r. 

In  Art.  105  it  was  shown  that  in  a  system  of  observation 
equations  the  p.  e.  r  of  an  observation  of  the  unit  of  weight  is 
found  from 


r=  0.6745 


y  n  —  Hi 


ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS      159 

where  [pt^]  is  the  sum  of  the  weighted  squares  of  the  residuals, 
V,  n  is  the  number  of  observation  equations  and  1:^  the  number 
of  independent  unknowns. 

Hence,  in  a  system  of  condition  equations,  ;/  being  the  num- 
ber of  observed  quantities  and  11^  the  number  of  conditions,  the 
number  of  independent  unknowns  is  n  —  n^,  and 

^   V  n  -  («  -  fi,) 

Luroth's  formula  (Art.  105)  may  be  used  as  a  check  on  the 
value  of  r. 

CJiccks  of  [/Tr]. — When  the  number  of  residuals  is  large, 
in  order  to  guard  against  mistakes  [/t'']  should  be  computed  in 
at  least  two  different  ways.  The  following  check  methods  will 
be  found  useful : 

(a)    The  correlate  equations  4,  Art.  1 19,  may  be  written, 

V/i7'i  =  ^u^a'c^  +  V^//c;  +  ■  •  • 


Square  and  add,  and 

[/>v^]  =  [uaa]  CjCj  +  2  [ua/?]  C^C^-\-  2  [iiac]  C^C^  -{-  ■  ■  • 

+     [ud/>]  C.,C,  +  2  [uifc]  QCs  +  ■  ■  . 

+     [uc^]  C3C3  4-  •  •  •        (2) 

+  •  •  • 
=  [C/]  from  (6),  Art.  119. 

(/?)  [/>7r-]  =  [C/] 


[uaa\       [uoo.i]  [//(V.2] 

+  r(/;^ +  /'":>■"  +  . 
+    .    .    .    . 


\!taa]       {/(/'/>.  \]        [urr.2] 
by  addition  attending  to  Eq.  [O,  Art.   I  19. 


i6o 


THE    ADJUSTMENT    OF    OBSERVATIONS 


This  expression  is  very  readily  computed  from  the  solution 
of  the  correlate  normal  equations,  as  shown  in  Ex.  2  following. 
Compare  the  computation  of  [vv]  from  the  scheme  in  Art.  io6. 

The  sum  [pv-]  can  in  general  be  computed  more  rapidly  by 
these  methods  than  by  the  direct  process  of  summing  the 
weighted  squares  of  the  residuals. 

122.  Ex.  I.  — The  three  angles  of  a  triangle  are  measured  with  the 
weights /j,  p.^,  ^3;  required  the  mean-square  error  of  a  single  observation. 

Using  the  values  of  Vi,v.,  v^,   found  in  Ex.  2,  Art.  120,  we  have 

r^  21      i/,I'«J'»J' 

_  /^ 
-[u] 


Hence 

Check  (i). 


M  = 


V[«]' 

[pv']  =  [cn 


smce  n,  =  i. 


as  before. 
Check  (2) 
Ex.  2.  — To  find  the  p.  e.  of  a  single  observation  in  Ex.  i,  Art.  120 


[pv^]  =  p^  directly  from  Eq.  3,  since  [7(aa]  =  i. 
\_u  J 


The  first  step  is  to  find  the  value  of  [pv-].     Three  methods  are  given  : 


(I) 


(2) 


p 

V 

pv" 

5 
7 
4 
7 
4 

—  0.05 

—  0.36 
+  0.68 

—  0.03 
-0.62 

.01 

.91 

1.85 

.01 

1.54 

4.32  =  {pv-'-X 

C 

/ 

CI 

-  0.230 

-  2.493 

—  0.76 

-  1.66 

0.17 
4.14 

4.31  =  [/''^] 

ADJUSTMENT    OF    COXDITIOXED    OBSERVATIONS      iGi 
(3)  From  tlie  solution  of  the  correlate  normal  equations  : 


c. 

a 

+  0.5929 

+  0.2500 

-  0.76  =  /' 

^ 

+  0.2500 

+  0.6429 

-  1.66   =   /  " 

► 

+  I 

+  0.4217 

-  1. 2818  =~   . 
[tc.aa] 

+  0.5375 

-  1-3395  =  /"•' 

1 

+  I 

-'•49^  "M.i] 

. 

.-.  [pv^  =  0.76  X  1.28 19  +  1.3395  X  2.492 
=  4-3136. 
Hence,  the  number  of  conditions  being  two, 

.  /4.31  „ 

r  =  0.6745  V/  "'^^  =  0.99  . 

123.    {b)  To  find  Up. 

Let  the  function  whose  weight  is  to  be  found  be 

F=f{V,,  K,...   V„), 

and  let  it  be  conditioned  by  the  11^  equations 


(4) 


(5) 


Expressing  F'\\\  terms  of  the  observed  values,  J/,,  J\r„,  .  .  .  J/,„ 
which  are  independent  of  one  another,  and  reducing  to  the 
linear  form,  we  have 

hF  8F^ 

8J/;  ^''  "^  SM, 
Hence,  as  in  Art.  108, 

/8/''V  /S/'^V  . 

"^  =  "^  [urj  +  "-•  [mj  + 

where  ti^,  u.^,  .  .  .  are  the  reciprocals  of  the  weights  of  the 
observed  values. 


(6) 
(7) 


Ex.  3.  —  To  find  the  m.  s.  e.  of  a  side,  a,  in  a  triangle  whose  angles  iiave 
been  measured  with  the  weights/,, /.,ji^3,  the  base,  b,  being  free  from  error. 


i62  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  function  equation  is 

„  ,  sin  A 

F  =  a  =  0  -. — JZ1 
sin  B 

and  the  condition  equation 

^  +  ^  +  C  =  i8o  +  6. 

Hence  from  Ex.  2,  Art.  120,  expressing  A,  B  in  terms  of  the  observed  values, 

A  =  M^^-  —  {180  +  e  -  {M^  +  M,  +  M^^, 

B  =  M.^+p-  {180  +  e  -  {M,  +  M.  +  3/3)}. 
Now, 

'^^  "  \hA  5Af,  +  dB  8M,r' 

/8F  SA        dF  SB^\  I8F  5A        dF  dB  \  _ 

"^  [dA  8M2  "^  5^  SMj"-  ^  [dA  dA/,      dB  SAI.f' 

=  .sin,"(|(,-j^,)cot^+^jCO.i;j,,, 

Therefore, 

«^=«^sin^i"(j(i  -^]cot^  +gcot^|%/, 

=  ^^sin^  I"   |(.,-^j)cot=^  +(;.,-g:)cot=5  +  ^^cot^cot5| 

and  M'jr  =  M  V«^, 

where  w  is  the  m.  s.  e.  of  a  single  observation. 

H  the  weights  ^„  p.,,  p^  are  each  equal  to  unity,  this  reduces  to 
u^  =  §  a}  sin^  \"  y?  (cot^  ^  +  cot^  5  +  cot  ^  cot  B), 

and  if  the  triangle  is  equilateral, 

V-F  =  3  '^^  sin^  i"  iit'. 
Also,  if  the  base,  instead  of  being  considered  exact,  had  the  m.  s.  e.  ms 
the  expressions  for  y.^  would  be  increased  by  -^  p-h-  and  y^  respectively. 

124.    It    is  usually  more   convenient   in  practice  to  use  the 
method  of  correlates. 

Let  the  function,  reduced  to  the  linear  form,  be  written 

dF=f'7',^f"v,^.    ■   ■  (l) 

This  is  conditioned  by  the  n^  equations,  also  in  the  linear  form, 


ADJUSTMENT    OF    COXDITIOXED    OBSERVATION'S      163 

a'i\  +  a"v.^  +  ...—/'  =  o, 

b'v^^b"i^'  +  .  .  .  -  l"=o,  (2) 


with 

[Z?'-]  =  a  minimum. 

Referring  to  the  principle  of  Art.  1 19,  we  see  tliat  by  using 
correlates  C^,  C.,,  .  .  .,  and  determining  them  properly,  we  can 
express  the  function  in  terms  of  the  quantities  7\,  z'.,,  .  .  .  v,,  as 
if  independent  ;  that  is, 

dF  ={f'  -a'C^-b'C-  .  ■  .)  i\ 

+  (^f"  -a"C,-b"C,-  •■■)v,+  ...  (3) 

and,  therefore, 

m,  =  (/'  -a'C^-b'C^-  .  .  .y  li^ 

+  (/"  -  a"C,  -  b"C,  -  ..  .y,,,-^  ...  (4) 

It  remains  to  determine  C^,  C„,  .  .  .  Now,  when  the  most 
probable  values  of  the  corrections  i\,  v„,  .  .  .  t'„  are  substituted 
in  the  value  of  the  function  dF,  this  function  must  have  its 
most  probable  value,  and,  therefore,  its  maximum  weight.  We 
may,  therefore,  determine  the  correlates  C  from  the  condition 
that  the  weight  of  dF  is  a  maximum ;  that  is,  that  nj,  is  a  mini- 
mum. Differentiate,  then,  //^  with  respect  to  C^,  Q  .  .  .  as 
independent  variables,  and  we  have  the  equations, 

[/ura]  C,  +  [uab]  C, -\-   ■  ■  ■   =  ['^'l/l 

[furb]  C\  +  [Nbb\  d+   .  .  .   =  [ubf\,  (s) 


from  which  C^,  C„...  arc  found. 

These  equations  being  precisely  of  the  form  of  ordinary  nor- 
mal equations,  it  follows,  as  in  {c)  and  {d).  Art  106,  that 

,,„.  =  luff-\  -  [uaf]  C,  -  [ubf]  C, (6) 

u,  =  wn  -  ^^  -  ^^mT]  —  ^^^ 

The  form  of  the  last  expression  for  ?//,.  shows  that  it  may  be 
found  by  means  of  the  following  scheme,  in  which  [uaf],  [nb/\ 
...  are  added  as  an  extra  column  in  the  solution  of  the  corre- 


164 


THE    ADJUSTMENT    OF    OBSERVATIONS 


late  normal  equations  (5),  in  the  manner  shown  in  /.rt.    106. 
For  three  correlates  the  scheme  would  be 


c. 

Q 

C3 

[uaa] 

[  uabl 
[  ubb] 

[  uac  ] 
[ubc] 
[  ucc  ] 

[  uaf] 
[  ubf] 

.... 

[ubb.i] 

[ubc.i  ] 

[  UCC-l  ] 

[ubf  a] 
[  ucf .  I  ] 

[»ff-^] 

.... 

[UCC.2'\ 

[ucf  .2] 

.... 

.... 

[^'ff}>\ 

=  np 

125.     Ex.  4.  —  To  find  the  weight  of  the  angle  PSB  in  Ex.  i,  Art.  118. 
Here  dF  =  —  t/j  +  v^, 

•■■/,=  -I,  /2  =  o,  /3  =  +I- 

From  the  condition  equations 

a'    =  +  I,  b''    =+  I, 

a"'  =  —  I,  b'"  =  —  I, 

a""=  +  I,  b""'=  +  I, 

.-.  [  uaf\  =  ^x-i  +  4x-i=-  0.45, 

Tlie  correlate  normal  equations  with  the  extra  column  for  finding  71  p : 


c,             c. 

/ 

+  0.5929 

-h  0.2500 
+  I 

+  0.2500 
+  0.6429 

+  0.4217 
+  0.5375 

+  I 

-  0.7600 

-  1.6600 

-  1.2818 

-  1-3395 

-  2.492 

—  0.4500  =  [uaf^ 
+  0.2500  =^  [ubf] 

—  0.7590 

—  0.0602  =  [ubfi] 

—  0.1120 

-r  0.4500  =  [uff] 

+  0.3416 

+  0.1084  =  [iffi] 

+  0.0067 

+  O.IOI7   =   [7iff.2] 
=   Up 

ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS      165 
Also  M/-  = 


=  1.47  V0.1017   from  Ex.  2 
=  0.47",  as  before. 
Ex.  5.  —  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  an  ang'e 
of  a  triangle  when  all  three  angles  are  measured,  their  weights  being /„/^^,,_^3, 
respectively. 

The  function  is  dF  =  7',, 

and  the  condition  equation      v^  +  v^  +  ■i';t  =  I- 

Hence  from  (15)  Up  ""  "*  ~  ivi 

_  u^  {u^  +  u^ 

Also  Mir  =  M  "^^F 

=  — =^i  /    ' (See  Ex.  i) 

V[«]  V  [«] 

The  weight  of  an  angle  before  adjustment  is  to  the  weight  after  adjustment 
as 

1:  [^^] 

or  til  +  «:i  ■•  [«]• 

II p^  =  ^,^  =  A  =  I'  the  weights  are  as  2  :  3.  This  result  is  independent 
of  the  magnitude  of  the  angle.  It  therefore  applies  to  any  problem  in  which 
the  condition  to  be  satisfied  is  that  the  sum  of  two  quantities  shall  be  equal 
to  a  third,  or  in  which  the  sum  of  all  three  is  equal  to  a  constant.  For 
other  solutions,  see  Ex.  2,  Art.  118. 

Ex.  6. —  If  n  angles  measured  at  a  station  close  the  horizon,  find  the 
weight  of  the  adjusted  value  of  any  one  of  them. 

[The  solution  is  exactly  as  in  the  preceding  example. 

The  weight  of  F„  for  instance,  is  found  from 

If  the  weights  /„  p,,  .  .  .  are  all  equal  to  one  another,  the  weight  of  an 
angle  after  adjustment  is  to  its  weight  before  adjustment  as 

;/:;/  —  !]. 
Ex.  7.  — Show  that  the  weight  of  the  sum  of  the  adjusted  angles  of  a 
triangle  is  infinite. 

[Sum  =  180  +  e,  a  fixed  quantity, 

.-.  p.  e.  =  o,  and  weight  =-  00, 

or  otherwise  ''1  +  '^'2  +  ^a  =  A 

a'  =  a"  =  a"'  =  i, 


i66 


THE    ADJUSTMENT    OF    OBSERVATIONS 


2'j?  =  3  - 


o]. 


Fig.S. 


Ex.  8.— In  the  "longitude  triangle"  Brest,  Greenwich,  Paris,  as  deter- 
mined by  the  U.  S.  Coast  Survey  in  1872,  the  observed  values  were 

>«.         J. 
Brest-Greenwich      ,     5i7-;7.i54i     weight  10, 
Greenwich-Paris      .      9     21.120,  "         7, 

Brest-Paris      ...     27     18.190,  "        9. 

Show  that  the  most  probable  values  are 

m.  s. 

17     57-i30>     weight  14 
9     21.086,  "        12 

27     18.216,  "        13 

126.  Ex.  9.  — To  find  the  weight  of  a  side  in  a  chain  of  triangles,  all 
of  the  angles  of  each  triangle  having  been  equally  well  measured  and  the 
base  being  free  from  error. 

Let  ^  be  the  measured  value  of  the  base,  and  let  a„  a.^,  .  .  .  an  be  the 
sides  of  continuation  in  order  as  computed  from  /; ;  a^  being  the  side  whose 
weight  is  required. 

If  A  ,  5„  A.„  B.„  .  .  are  the  measured  values  of  the  angles  used  in 
computing  al  from  d,  the  angles  A^,  A.„  .  .  .  being  opposite  to  the  sides  of 
continuation,  then 

a^_smA,       «:,       sin  ^^  ^n    _sin^„ 

J~ 


sin^i'     «i       sin  ^2'  '  «n-i       sin  ^„ 

Hence,  by  multiplying  these  expressions  together, 

sin  An 


sin^,    sin^o 
an  =  o  —. — ^^    —. — —^ 
sm  j5i    sm  B^ 


sin  En. 


(I) 


We  may  now  proceed  in  two  ways. 

{a)  Differentiating  directly, 

dan  =  an  sin  i"[cot  A  {A)  -  zoK.  B  (5)], 
where  {A)^  (B),  .  .  .  denote  the  corrections  to  A,  B,  .  .  . 

[In  a  chain  of  triangles  it  is  convenient  to  use  the  notation  (^),  (B),  . 
for  7/„  7A„  .  .  .  the  parentheses  indicating  corrections.] 

The  condition  equations,  from  the  closure  of  the  triangles,  are 

(A,)  +  (B,)  +  (Q  =  /', 

(A,)  +  (5,)  +  (Q)  =  /",  (2) 


Substituting  in  Eq.  7,  Art.  124, 

«a»  =  3  ^n  sin'  i"  [cot^  A  +  cot^  B  +  cotA  cot  B], 
the  result  required. 

If  the  triangles  are  equilateral,  this  reduces  to 
«     =  2  nP  sin^  i". 


(3) 


(4) 


ADJUSTMENT    OF    COXDITIOXED    OBSERVATIOXS      167 

Hence  in  a  chain  of  equilateral  triangles  the  weights  of  the  sides  decrease 
as  we  proceed  from  the  base,  b,  through  the  successive  triangles,  inversely 
as  the  number  of  triangles  passed  over ;  that  is,  are  as  the  fractions 


{b)  Taking  logs  of  both  members  of  Eq.  i,  and  differentiating, 
"^  ^°^  "^'^  ""  dA   '°^  ^'"  ^'  '^^^  ~  ~dB  '°^  ^'"^  ^'  ^^^  +  •  •  • 

or  expanding  the  first  member, 

where  5^  is  the  tabular  difference  for  one  unit  for  the  number  a^^,  and  5  j,  5^.. 
are  the  logarithmic  differences  corresponding  to  i"  for  the  angles^,  B,  in  a 
table  of  log  sines. 

Hence,  attending  to  the  condition  equations  2,  we  have  from  (7)  Art.  124 
for  Eq.  5,  u^^^  „„  =  5  [5^/  +  5  ^  5^,  +  5^=] , 

and  for  Eq.  6,  tcan.  =  "  5—;,  [5  .^  +  5,5,,  +  5  ,.=], 

as  giving  the  weight  of  the  logarithm  of  the  side  and  ihe  weight  of  the  side 
respectively. 

Of  the  two  forms  (a)  and  (^),  the  logarithmic  is  in  general  the  more  con. 
venient  in  practice. 

Compare  the  final  forms  derived  here  with  those  shown  in  Ex.  3,  p.  164. 

Ex.  10.  —  From     a     base  b        n 

AB  (=  b)  proceeds  a  chain  /V      TV        /V,     bJ^V       "/V      T 

of   equilateral    triangles,    all  /     \  /     \  /     \cy    '  \  /      \  /  ^ 

of  the  angles  being  equally  ^7 ^ ^ ^^-^ '-^ ^ 

well  measured,  and  the  sides  Pip  6. 

BC,  CD,    .  .  .  being  in  the  same  straight  line.     Find  the  m.  s.  e.  of  the  line 

jffTV,  which  is  n  times  the  base. 

Take  first  the  simple  case  of  n  =  2. 

„       „,,.       .  sin  C.    ,    ,  sin  >f.  sin  ^.,  sin  (7, 
/'  =  BN  =  b  ~. — ^'  +  b  -. — j^. — jy^. — jj- 
sin  Bi  sm  i?,  sm  B.  sm  B^ 

.-.  d/^  =  {cot  AiiAi)  -2  cot  Bi  {B,)  +  cot  C,  (C,) 

+  cot  A-i  (A 2)  -     cot  Bn  (Bj) 

-     cot  B3  (B-J  +  cot  C3  {C.j)}b  sin  i". 

Also,  we  have  the  condition  equations 

(A,)  +  (B,)  +  (C,)  =  /'. 
(A,)  +  (B,)  +  (€,)  =  /", 
{A,)  +  (B,)  +  (Q  =  /'". 


i68 
Hence 


THE    ADJUSTMENT    OF    OBSERVATIONS 


[aa]    =  3,  [af]   =  o, 

[CC.2]  =  3,  [Cf.2]  =  O. 

\J/'\==icof'  Ai  +  4Cof  Bi  +  cot^  Ci  +  cot^  A^  +  cot^  Br,  +  cof^  Bs+cot^  Q)  P  siri'  i' 
=  J^  P  sin*  i",  since  cot^  60°  =  ^. 
Substituting  in  Eq.  7,        ^£]^=  V  '^^  sin^  i", 
and  therefore,  M^jV  =  A^  V  V'  '^  sin  i", 

where  m  is  the  m.  s.  e.  of  an  observed  angle. 
Generally, 

dF  =  {n  -  \)  cot  A,{A^)  -  ti  cot  B^  {B,)  +  cot  Ci(C,) 
+  («  -  i)  cot  ^2  (^2)  -  («  -  0  coiBoiB^) 

-  («  -  I)  cot  ^3  (.93)  +  cot  C3  (Q 


+ 


and 


,,   .  ,    ,,(4  n^  —  3«'  +  5  ;A 
=  ^2  sin*  I    I  ^^^ ~ ^—  1 


If  the  chain  proceeds  in  the  opposite  direction  until  AN'=  BN,  then, 
since  /*^^v/^=  /^^^v'^,  and  NN'=  2  /^//  approximately,  we  have 

/^AW'^=  /^^^'  s>n  I'Y' ^, ^- 

If  NN'  is  11  times  the  base  (putting  «  =  -  J 

Maw  =  /^^VA^'  sin  i"y :tit 

Hence  it  follows  that  in  a  chain  of  equilateral  triangles  where  one  base  only 
is  measured,  it  is  better  to  place  the  base  at  the  center  of  the  chain  rather 
than  at  either  end. 

Ex.  II.  —  If  two  similar  isosceles  triangles  on  opposite  sides  of  the  base 
.<4Care  measured  independently,  thus  forming  a  rhombus  (vertices  B,  B'), 
then,  taking  the  weight  of  each  angle  unity. 


P-RR'   = 


J^b  sin  i"  ,  B 
^^ —  cosec-  — 


24 


B 


and  if  BB'  is  n  times  the  base  h,  then,  since  cot  —  =  n, 

t^BB'  sin  i' 


I^BB' 


2\/3 


nl 


Caution.  —  If  we  solved  for  the   rhombus  directly, 
it  would  not  do  to  take 


BB'=  b  cot 


B 


and  then  form  M^^,.  The  result  would  be  \^2  times  too  great.  For  as  the 
triangles  are  measured  independently,  each  half  of  BB'  must  be  considered 
separately,  so  that  we  must  use  the  form 


ADJUSTMEN'T    OF    COXDITIOXED    OBSERVATIONS      169 


BB'  =  -  (  cot  --  +  cot 


with  the  condition  equations 

(^)  +  (5)  +  (CC)  =  /,, 

(^')  +  (5')+  (CC')  =  4, 
corresponding  to  the  angles  of  the  two  triangles. 

Solution  in   Two   Groups. 

127.  In  geodetic  work  it  often  happens  that  the  observed 
quantities  are  subject  to  a  simple  set  of  conditions  which  may 
be  readily  solved  as  observation  equations  by  the  method  of 
independent  unknowns,  and  are  also  subject  to  other  conditions 
which  are  best  solved  by  the  method  of  correlates.  The  equa- 
tions are  thus  divided  into  two  groups  for  solution,  and  the  com- 
plete solution,  therefore,  consists  of  two  parts.  The  observation 
equations  forming  the  first  group  are  solved  by  themselves  and 
give  approximations  to  the  final  values  of  the  unknowns.  The 
corrections  to  these  approximate  values  due  to  the  second 
group  are  next  found  by  solving  this  second  group  by  the 
method  of  correlates. 

The  merit  of  the  method  consists  in  utilizing  the  work  ex- 
pended in  the  solution  of  the  first  group  in  determining  the 
additional  corrections  due  to  the  second  group.  The  solution 
is  rigorous,  and,  being  broken  into  two  parts,  is  more  easily 
managed  than  if  all  the  equations  had  been  solved  simultaneously. 

Let  the  first  group  of  equations  be  the  observation  equations, 
n  in  number  and  containing  ;/„  unknowns  (;/  >  ;/„), 

a^x  ■-(-  h,y  -j-  ...-/,  =  z'l  ,  weight/,, 

a.,x  +  b.j  4-  ■  •■-/,  =  ?'. ,        "      Pv  (i) 


and  the   second  group  the  condition  equations,  n^  in  number, 
involving  the  same  unknowns  (n^  <  n„), 

a'x  +  a" y  -f  •  •  ■  —  /'  =  o, 

b'x  -f  b"y  +...-/"  =  o,  (2) 


170  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  most  probable  values  of  the  unknowns  x,  j,  .  .  .  are  those 
which  are  given  by  the  relation 

[^z;-]  =  a  minimum.  (3) 

It  is  required  to  find  them. 

The  value  of  an  unknown  is  found  in  two  parts,  the  first,  (.i'), 
(j),  .  .  .  arising  from  the  observation  equations,  and  the  sec- 
ond, (i),  (2),  .  .  .  arising  from  the  condition  equations,  thus: 

X  =  (x)  -h  (i), 

_^=(j)+(2),  (4) 

Now,  overlooking  for  the  present  the  condition  equations,  and 
taking  the  observation  equations  only,  (.v),  (j),  .  .  .  would  be 
found  by  solving  these  equations  in  the  usual  way.  We  have, 
therefore,  reducing  all  to  weight  unity  for  convenience  in  writing, 
the  normal  equations 

[aa]  (x)  +  [.^/;]  (j)  +  .  .  .  =  [a/], 

H](^-)  +  [/V.]0')  +  ...  =[l>/],  (5) 


The  solution  of  these  equations  gives  (see  Art.  103) 
(x)  =  [cm]  [a/]  +  [a/3]  [/>/]  +  •  •  • 

(y)  =  mM  +  [mm  +  •••  (6) 


Hence  (x),  (j),  .   .  .  are  known. 

To  find  the  condition  corrections  (i),  (2),  .  .  .  ,  eliminate  7\, 
v^,  .  .  .  v^  by  substituting  in  the  minimum  equation,  which  then 
becomes, 

[aa]  XX  -\-  2  [a/>]  xy  +  •  •  ■  —  2  [al]  x, 

+      [hl>]yy+  •■■  -  2[bl]x,  (7) 


+  [//]  =  a  min. 

This  equation  is  conditioned  by  equations  2.     Thus,  the  solution 
is  reduced  to  that  already  carried  out  in  Art.  1 19. 

Calling  /,  //,...  the  correlates  of  equations  2,  we  have  the 
correlate  equations 


ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS      171 

[aa]x  +  [(7^]^  +  •  •  •  -  [«/]  =  a  /  +  l>'  //  +  ■  ■  ■ 
[ad\x  +  L'^^j^  +  •  •  •  -  [/'/J  =  <^"^  +  ^"^^  +  •  •  • 

These  equations,  taken  with  (4)  and  (5),  give  the  relations 

[aa\  (1)  +  [al^\  {2)  +  ■  •  •  =  a  I  ^  b'  JI  ^  ...  =[7]  suppose 
\ab\  Ky)  +  \bb\  (2)  +   .  .  •   =  a"/+  b"II^  •  •  •  =0.       "       (8) 

which  being  of  the  same  form  as  (5),  their  sokition  gives 

(l)  =  [aa]|T]  +  [a^]g+    ••• 

(2)  =  [a^]  0  +  \m  0  +  •  •  •  (9) 

or  substituting  f or  0] ,   [2] ,  .  .   .  their  vahies  from  (8), 

(i)  =  a'  /+  p/  //+  c'  ///+  •  •  • 

(2)  =  a'V+  b"//+  c"///+  .  .  .  (10) 

where 

A'  =  [aa]  a'   4-  [aiS]  «"   +    ••• 

B'  =  [oa]  b'  +  [ai8]  /^"    +    •  •  .  (11) 

and  are  known  quantities. 

We  have,  therefore,  expressed  the  corrections  (i),  (2),  .  .  . 
in  terms  of  the  unknown  correlates,  /,  //,...  It  remains  now 
to  find  these  correlates. 

Substituting  for  x,  y,  .  .  ■  their  values  from  (4)  in  the  con- 
dition equations,  and 

a'  (i)  -h  rt-"  (2)  +  •  •  •  =  C. 

b'  (i)  +  /^"  (2)  +  ■  •  •  =  K',  (") 

where 

/,;   =  /'   -a'  {X)  -  a"  (y) 

/;'  =  /"  -  b'  (x)  -  b"  (y)  -  •  •  •  (;3) 

and   are,    therefore,  known   (luantities,   since    (.r),    (y),  •  •   •  arc 
known. 


172  THE    ADJUSTMENT    OF    OBSERVATIONS 

Substitute  the  values  of  (i),  (2),  .   .  .  from  (10)  in  (12),  and 
we  have  the  correlate  normal  equations, 

[^]  7+  [^]  11+  .  .  ■  =/J, 

[^]  /+  [^]  //+  ...  -  /o",  (14) 


where 

[^aJ    =  [aa](r'i/     +  [a/3  ]</'</'    +    ■   .    . 

+  [a/3J  yy' +  [/3y8j  ./V  +   ...  (15) 

The  solution  of  equations  14  gives  the  correlates  /,  //,... 
Hence  the  corrections  (i),  (2),  .  ,  .  are  known.  Also,  since 
(x),  (j),  .  .  .  have  been  found  from  (6),  the  total  corrections 
X,  y,  .   .  .  are  known. 

128.  In  carrying  the  preceding  solution  into  practice,  the 
following  order  of  procedure  will  be  found  convenient : 

(a)  The  formation  and  solution  of  the  observation  equa- 
tions (i). 

The  partially  adjusted  resulting  values  (.r),  (y),  .  .  .  are  now 
to  be  used. 

(d)    The  formation  of  the  condition  equations  (12). 

a'  (i)  + ./'  (2)  +  . . .  =  /;, 


(c)  The  formation  of  the  weight  equations  (9).  They  are 
at  once  written  down  from  the  general  solution  of  the  observa- 
tion equations  in  (cr),  and  are 

(l)  =  [aa]   g   +  [a^]  0  +  •    •   • 

(2)  =  [a/3]  0  +  im  a  +  •  •  • 


(d)    The  formation  of  the  correlate  equations  (8). 
g  =  y  /  +  />'//  +  ■  ■  ■ 


ADJUSTMENT    OF    CONDITIONED  OBSERVATIONS        173 

(e)    The  expression  of  the  corrections  in  terms  of  the  corre- 
lates by  substituting  from  (d)  in  (r). 

(i)  =  a'/  +  hV/  +  .  .  . 
(2)  =  a"7+  b"//+  .  .  . 


(/)    The  formation  of  the  normal  equations  by  substituting 
from  (e)  in  (d).     They  are, 

[^]/+  [.7i]//+  . .  .  =/;, 


(^)  The  determination  of  the  corrections  by  substituting 
the  values  of  the  correlates  in  (e). 

129.  To  Find  the  Precision  of  the  Adjusted  Values  or  of 
any  Function  of  them. 

(a)    First  find  r,  the  p.  e.  of  an  observation  of  weight  unity. 

We  have  (Art.  121)  r  =  .6745  fi, 


IX-  = 


number  of  conditions 


{n  —  ;/„)  +  ;/<- ' 

since  n  —  n^  is  the  number  of  conditions   in  the   observation 
equations,  and  n^  the  number  in  the  condition  equations. 
To  find  \y'-\     From  the  first  observation  equation 

z\  =  a^x  +  b^y  +  •  •  •   —  /j 

=  ci,  {x)  +  b,  ( j)  +  .  .  .  -  /^  4-  ^1  (i)  +  ^.  (2)  -H  •  .  . 

=  V  +  ^l(0   +  '^l(2)+    ••    • 

Similarly 


where 

v^  =  a^{x)  ^  b,{y)  +•••-/!, 
v^  =  a^{x)  ^  b„Xy)  ^  •  •  •  -4, 


174  THE    ADJUSTMENT    OF    OBSERVATIONS 

that  is,  v^,  v^,  .  .  .  are  the  residuals  arising  from  taking  the 
observation  equations  only. 

Attending  to  Eq.  5,  Art.  1 27,  it  follows  evidently  that 

[av''\  =  o  [/w"]  =  o,  .  .  . 

Square  the  residuals  v^,  v^,  .  .   .  and  add,  then 

[7-2]    =    [^,0,,0]    +   [^^  (i)    +    /,  (3)    ^_     .    .    .     |2J 

=  [vh'^]  +  [ww]  suppose. 

The  total  sum  [zr]  may  therefore  be  found  in  two  parts,  one 
from  squaring  the  residuals  of  the  observation  equations,  and 
the  other  from  the  corrections  (i),  (2),  .  .  . 

We  proceed   to  put   [zuxa]   in  a   more    convenient    shape  for 
computation. 

[WW]  =  [\a{i)  +  ^(2)  +  .  .  .  |2] 

=  {i)\{aa]{i)  +  [ab]{2)+  ..-] 
+  {2)\[ab]{i)  +  [bl>]{2)+  ...  I 

+ 

=  (i)  [7]  +  (2)  [7]  +  .  .  . 

from  Eq.  8,  Art.  127. 

Substitute  for  (i),   [T|,   (2),  .  .  .  their  values  from  equations 
8  and  10,  Art.  127,  and  expand;  then 

[ww]  =  \  [«a]  /  +   [ob]  11+  ...   )  / 

+  [[^]/+  [^]  //+  .  ■  ■  \  II 
+ 

which  may  be  transformed,  by  means  of  Eq.  14,  into  the  form 

[ww]  =1^1+  i^' 11+  ... 
or,  as  in  Art.  121,  into  the  form 

[(^jiaJ        [Z-b.  i]        [^c.2] 

These  forms  may  be  readily  computed  as  in  Art.  106.  , 

130.    {b)    Next  find  the  weight  of  the  given  function  of  the 
adjusted  values. 


ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS      175 

Let  the  function,  reduced  to  the  linear  form,  be 

dF  =  -^.v  +  g.y  +  .  .  .  (16) 

where  g^,  gn,   .  .   .  are  known  quantities. 

Put  for  ,r,  J,  .  .  .  their  values  {x)  +  (i),  {y)  +  (2),  .  .   .  and 

Put  for  (i),  (2),  .   .  .  their  values  from  (10),  and 

dF=g^  (x)  +  g,  0)  +   .  .  .  +  [.A]  /+  [^B]  11+  .  •  .     (ij) 
where  /,  //,...  are  found  from  the  equations 
[^  /+  l^]  //+  ...-/;=  o, 

[^]  i+[Jf\  //-\- /;'=  o, 

Using  the  muhipliers  A-^,  k^,  .  .  .  in  order  to  elmimate  /,  //, 
.  .  .  ,  we  have,  as  in  Art.  121, 

dF=  g^  ix)  +  ^-,  (jO  +  •  •  •  +  /,;^',  +  hl'k,  +  ■  •  • 
+  {  [^-a]  -  [^]  k,  -  [^]  k,-  .  .  .)/, 

+  I  M  -  [^«]  ^1  -  [^]4 J// 

+ (18) 

We  may  determine  k^,  k^_,  .  .  .  so  as  to  cause  the  coeffi- 
cients of  /,  II,  .  .  .  to  vanish  ;  that  is,  so  as  to  satisfy  the 
equations 

[^a]  k^  +  [.?k]  k.^+  .  .  .  =  [gA], 

[^]  k,  +  [/;?.]  k,  +  .  .  .  =  [g^l  (,9) 

and  then  we  shall  have 

dF  =  g,  {x)  +  g,  {y)  +  .  .  .  +  i:k,  +  hl'k,  +  •  •  • 

Substitute  for  I',  /,/',  .   .   .  from  (13),  and 

dF=  \lk-\  +  G,  (a)  +  G.,  (,v)  +  ■  •  •  '  (20) 

where 

<?!  =  A  -  ^'k\  -  ^''h  -  ■  '  ■ 

G,=g,-  a"k,-/>"/,.^ (2,) 


176 


THE    ADJUSTMENT    OF    OBSERVATIONS 


We  have  thus  expressed  the  function  in  terms  of  (x),  (y),  .   .  . 
and  known  quantities. 

Now,  since  (,r),  (j),  .   .  .  are  not  independent,  but  are  con- 
nected by  the  equations 

[aa]  {x)  +  [ab]  {})  +  •  •  .  =  [al\ 


the  problem  is  reduced  to  that  already  solved  in  Art.  108. 
If,  therefore,  Uj,.  is  the  reciprocal  of  the  required  weight, 

uf  =  \GQ\  (22) 


where 


<2,  =  [a/3]  G,  +  \m  G.  + 


(23) 


the   quantities   [aa],   [a/3],  .   .  .  being  as   in  the   weight   equa- 
tions 9. 

Putting  for   G^,   G.-,,  .   ,  .   their    values    from    (21)    in   these 
equations,  and  attending  to  (u),  we  find 

Qt  =  ^1  ~  a'  ^1  ~  b'  '^'2  —  •  •  • 


where 


Substituting  in  (22)  for  G^,  G,^,  .  .   .   Q^,  Q„, 
from  (21)  and  (24), 

[GQ]  =  [gq]  -     [gA ]  k,  -[gB]  k,  - 


+ 


(24) 

(25) 

their  values 


But  from  (11)  and  (25) 

\fi4\  =  U^\  [h'\  =  ls^\  ■  ■  • 
Hence,  attending  to  (19),  the  above  expression  reduces  to 


ADJUSTMENT    OF    CONDITIONED    OBSERVATIONS       177 

[c;<2]  =  U^^]  -  U-A]  X',  -  [,-B]  i', 

or  to 

To  compute  [£:q].     Multiply  each  of  equations  25,  in  order,  In- 
^1.  £'■2'  •  •  •  »  ^'^"<^  ^^^^^'  ^""^1 

[iV]  =  [""]  ^ii\  +  ~  [«/3]  S,g-i  +  •  •  • 

+  ■  •  • 
where  \aa\  [a/3],  .  .   .  may  be  taken  from  the  weight  equations. 
The  remaining  terms  of  the  second  form  of  \GQ\  may  be 
found  from  the  solution  of  the  normal  equations,  as  shown  in 
Art.  121. 

Solution  by  Successive  Approximation. 

131.  This  method  of  solution  (due  to  Gauss)  is  of  the  great- 
est importance  in  adjustments  involving  many  conditions.  It 
may  be  stated  as  follows  : 

The  condition  equations  may  be  divided  into  groups,  and 
the  groups  solved  in  any  order  we  please.  Each  successive 
group  will  give  corrections  to  the  values  furnished  by  the  pre- 
ceding groups,  and  the  corrected  values  will  be  closer  and 
closer  approximations  to  the  most  probable  values  which  would 
be  found  from  the  simultaneous  solution  of  all  the  groups. 

For  suppose  we  have  the  condition  equations 


with 


a' 

V,  +  a"v. 

+ 

L 

b' 

V,  +  b"v.. 

+ 

■  = 

4, 

h' 

V,  +  h"v. 

+ 

.  = 

//., 

k' 

l\   +  k"T, 

+  ■ 

•     = 

4, 

\pv-\ 

=  a 

niin. 

be 

the  valu 

es 

of 

'•^v 

^'2' 

Let  z//,  v^,  ...  be  the  values  of  v^,  v„,  .  .  .  obtained  from 
solving  the  first  group  alone  ;  that  is,  from 


178  THE    ADJUSTMENT    OF    OBSERVATIONS 

a'i\    +  a"v^  +  •  •  •  =  4, 
b'vl  +  b"v^  +  .  .  .  =  4, 


\pv"^\  =  a  min. 

If  now  (7'/),  (t'Z),   ...  are  the  corrections  to  these  values  re- 
suhing  from  the  remaining  equations,  then  since 

=  [/>zr-]  +  [p(z/y], 
the  condition  equations  are  reduced  to 


/^'M  +^"(r./)  +  •  •  -  =  4', 

with 

[/  (^0']  =  a  min., 

and  the  values  of  (v')  found  from  the  simultaneous  solution  of 
these  equations,  added  to  the  values  of  v'  found  from  the  solu- 
tion of  the  first  set,  would  be  equal  to  the  value  of  v  found 
directly. 

Similarly,  if  v/',  <',  ...  be  the  values  of  (?/)  obtained  by 
solving  the  second  set  alone,  and  {i\"),  {v^"),  ...  be  the  cor- 
rections to  these  values  resulting  from  the  remaining  equa- 
tions, then  since 

[pir-\  =  {pv'^-]  +  [p7'"'-]  +  [p{2/J] 

the  condition  equations  are  reduced  to 

a'(0  +  ^"«)+  •  •  •  =  ''«" 
t'  (,,/')  +  //'  (e,/')  +  .  .  .  =  4", 

//(0  +  ^^"(0+--  -  =  4", 
k' {v;')  +  k"  {zu!')  +  ■  ••  =  4", 

with 

[/  {^'"y]  =  a  min. 


ADJUSTMENT    OF   CONDITIONED    OBSERVATIONS      179 

The  quantities  [/c/-],  [/:'"-],  .  .  .  being  positive,  the  minimum 
equation  is  reduced  with  the  solution  of  each  set,  and  thus  we 
gradually  approach  the  most  probable  set  of  values.  Beginning 
with  the  first  set  a  second  time,  and  solving  through  again,  we 
should  reduce  the  minimum  e4uation  still  farther,  and  by  con- 
tinuing the  process  we  shall  finally  reach  the  same  result  as 
that  obtained  from  the  rigorous  solution.  In  practice  the  first 
approximation  is  in  general  close  enough. 

It  is  plain  that  the  most  probable  values  can  be  found  after 
any  approximation  by  solving  simultaneously  the  whole  of  the 
groups,  using  the  values  already  found  as  approximations  to 
these  most  probable  values. 

Examples  will  be  found  in  the  next  chapter. 


CHAPTER  VI 

APPLICATION      TO       THE      ADJUSTMENT      OF     A      TRIANGULATION. 
METHOD    OF    ANGLES 

132.  The  adjustment  of  the  measured  angles  of  a  triangula- 
tion  net  is  a  special  case  of  the  problem  discussed  in  the 
preceding  chapters.  We  assume  the  reader  to  be  acquainted 
with  the  construction  and  method  of  handling  of  instruments 
used  in  measuring  horizontal  angles,  and  shall  confine  ourselves 
to  the  methods  of  adjusting  the  measured  values  of  the  angles. 

133.  For  clearness  we  will  explain  in  some  detail  the  pre- 
liminary work  necessary  for  the  formation  of  the  condition 
equations.  In  a  triangulation  there  must  be  one  measured  base 
at  least,  as  AB.     Starting  from  this  base,  and  measuring  the 

angles  CAB,  ABC,  we  may  compute  the  sides, 
AC,  BC  by  the  ordinary  rules  of  trigonometry. 
In  plotting  the  figure,  the  point  (Tcan  be  located 
in  but  one  way,  as  only  the  measurements 
necessary  for  this  purpose  have  been  made. 
Fig.  8.  Similarly,  by  measuring  the  angles  CBD,  DCB 

we  may  plot  the  position  of  the  point  D,  and  this  can  be  done 
in  but  one  way.  If,  however,  the  observer,  while  at  A,  had  also 
read  the  angle  DAB,  then  the  point  D  could  have  been  plotted 
in  two  ways,  and  we  should  find  in  almost  all  cases  that  the  lines 
AD,  BD,  CD  would  not  intersect  in  the  same  point.  In  other 
words,  in  computing  the  length  of  a  side  from  the  base  we  should 
find  different  values,  accordins:  to  the  triangles  through  which  we 
passed.  Thus  the  value  of  CD  computed  from  AB  would  not, 
in  general,  be  the  same  if  found  from  the  triangles  ABC,  BCD, 
and  from  ABC,  CAD. 

180 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        iSi 

If  the  exterior  angle  ABD  had  also  been  measured,  we  should 
have  another  contradiction,  arising  from  the  non-satisfaction  of 
the  relation 

BBC  +  CBA  +  ABD  =  360°. 

And  not  these  contradictions  only.  For  we  have  considered 
so  far  that  in  a  triangle,  only  two  of  the  angles  are  measured. 
If  in  the  first  triangle,  ABC,  the  third  angle,  BCA,  were  also 
measured,  we  know  from  spherical  geometry  that  the  three 
angles  should  satisfy  the  relation 

CAB  +  ABC  -+■  BCA  =  180°  +  sph.  excess  of  triangle, 

which  the  measured  values  will  not  do  in  general.  A  similar 
discrepancy  may  be  expected  in  the  other  triangles. 

In  a  triangulation  net,  then,  with  a  single  measured  base,  in 
which  the  sides  are  to  be  computed  from  this  base  through  the 
intervening  triangles,  we  conclude  that  the  contradictions  among 
the  measured  angles  may  be  removed  and  a  consistent  figure 
obtained  if  the  angles  are  adjusted  so  as  to  satisfy  the  two 
classes  of  conditions  : 

(i)  Those  arising  at  each  station  from  the  relations  of  the 
angles  to  one  another  at  that  station. 
These  are  known  as  /oca/  conditions. 

(2)  Those  arising  from  the  geometrical  relations  necessary  to 
form  a  closed  figure. 

(a)  That  the  sum  of  the  angles  of  each  triangle  in  the  figure 
should  be  equal  to  180^  increased  by  the  spherical  excess  of  the 
triangle. 

(d)  That  the  length  of  any  side,  as  computed  from  the  base, 
should  be  the  same  whatever  route  is  chosen. 
These  are  known  as  genera/  conditions. 

134.  The  number  of  conditions  to  be  satisfied  will  depend 
on  the  measurements  made.  Each  condition  can  be  stated  in 
the  form  of  an  equation  in  which  the  most  j^robable  values  of 
the  measured  quantities  are  the  unknowns.  The  numlx-r  of 
equations  being  less  than  the  number  of  unknowns,  an  infinite 


i82  THE    ADJUSTMENT    OF    OBSERVATIONS 

number  of  solutions  is  possible.  The  problem  before  us  is  to 
select  the  most  probable  values  from  this  infinite  number  of 
possible  values. 

The  general  statement  of  the  method  of  solution  is  this. 
Adjust  the  angles  so  as  to  satisfy  simultaneously  the  local  and 
general  conditions  ;  that  is,  of  all  possible  systems  of  corrections 
to  the  observed  quantities  which  satisfy  these  conditions,  to  find 
that  system  which  makes  the  sum  of  the  squares  of  the  correc- 
tions a  minimum. 

The  form  of  the  reduction  depends  on  the  methods  employed  in 
making  the  observations.  These  methods,  in 
general  terms,  are  as  follows  :  Let  O  in  the 
figure  be  the  station  occupied,  and  A,  B,  C 
signals  sighted  at.  The  angles  AOB,  BOC 
are  required.  By  pointing  at  A  and  then  at 
B  we  find  the  angle  AOB.  Point  now  at  B 
^'2- 9-  and  next  at  C,  and  we  have  the  angle  BOC. 

These  two  angles  are  indepoideiit  of  one  another. 

If,  however,  we  had  pointed  at  A,  B,  Cm.  succession  we  should 
also  have  found  the  angles  AOB,  BOC,  but  they  would  not  be 
independent  of  one  another,  as  the  reading  to  B  enters  into 
each. 

The  first  method  of  measurement  is  known  as  the  method  of 
independently  measured  angles,  or,  if  each  angle  is  mechanically 
multipHed  before  being  read  off  from  the  circle,  it  is  called  the 
method  of  repetition  ;  and  the  second  method  is  known  as  the 
method  of  directions. 

The  Method  of  Independent  Angles. 

135.  As  the  case  of  independent  angles  is  the  simplest  to 
reduce,  we  shall  begin  with  it. 

A  distinction  must  be  made  between  angles  that  are  inde- 
pendently observed  and  angles  which  are  independent  in  the 
sense  that  no  condition  exists  between  them.      Thus  at  the 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        183 

station  O,  above,  the  angles  ^(9^5,  BOC,  AOC  might  be  observed 
independently  of  one  another,  but  we  should  not  call  them 
independent  angles,  since  the  condition 

AOC=  AOB  +  BOC 

must  be  satisfied  between  them.  By  independent  angles, 
therefore,  in  the  reduction,  we  mean  those  measured  angles  in 
terms  of  which  all  the  measured  angles  can  be  expressed  by 
means  of  the  conditions  connecting  them.  In  the  present  case 
any  two  of  the  three  angles  AOB,  BOC,  AOC  may  be  taken  as 
independent,  and  the  third  angle  would  be  dependent. 

Angles  may  be  measured  independently  either  with  a  repeat- 
ing or  with  a  non-repeating  theodolite.  In  primary  work  a  non- 
repeating theodolite  in  which  the  graduated  limb  is  read  by 
microscopes  furnished  with  micrometers  is  to  be  preferred. 
The  method  of  reading  an  angle  is  as  follows:  The  instrument, 
having  been  carefully  adjusted,  is  directed  to  the  left-hand  signal 
and  the  micrometers  read.  It  is  then  directed  to  the  other 
signal  and  the  micrometers  again  read.  The  difference  between 
these  readings  is  called  a  positive  single  result.  The  whole 
operation  is  repeated  in  reverse  order ;  that  is,  beginning  with 
the  second  signal  and  ending  with  the  first,  giving  a  negative 
sinde  result.  The  mean  of  these  two  results  is  called  a  com- 
bined  result,  and  is  free  from  the  error  arising  from  uniform 
twisting  of  the  post  or  tripod  on  which  the  instrument  is  placed, 
or  from  "twist  of  station,"  as  it  is  called,  provided  the  rate  of 
observing  is  constant. 

The  telescope  is  next  turned  180°  in  azimuth  and  then  180°  in 
altitude,  leaving  the  same  pivots  in  the  same  wyes,  and  another 
combined  result  is  obtained.  The  mean  of  the  two  coiiihined 
results  is  free  from  errors  of  the  instrument  arising  from  imper- 
fect adjustments  for  collimation,  from  inequality  in  the  heights 
of  the  wyes,  and  from  inequality  of  the  pivots. 

The  distinction  between  these  two  combined  results  is  noted 
in  the  record  by  "telescope  direct  "  and  "telescope  reverse." 


i84  THE    ADJUSTMENT    OF    OBSERVATIONS 

136.  Besides  those  mentioned,  there  are  two  kinds  of  system- 
atic error  in  measuring  angles  that  deserve  special  attention. 
They  are  the  errors  arising  from  the  regular  or  "periodic" 
errors  of  graduation  of  the  horizontal  limb  of  the  instrument, 
and  the  error  from  the  inclination  of  the  limb  itself  to  the  horizon. 
The  effects  of  the  first  may  be  got  rid  of  by  the  method  of  ob- 
servation, as  follows:  The  reading  of  the  limb  on  the  first  signal 
is  changed  (usually  after  each  pair  of  combined  results)  by  some 
aliquot  part  of  the  distance,  or  half-distance,  between  consecu- 
tive microscopes  in  case  of  two-microscope  and  three-microscope 
instruments  respectively.     Thus,  if  n  is  the  number  of  pairs  of 

combined  results  desired,  the  changes  would  be and  re- 

n  71 

spectively  with  the  instruments  mentioned.  The  operation  of 
reversal  in  case  of  a  three-microscope  instru- 
ment causes  each  microscope  to  fall  at  the 
middle  of  the  opposite  120°  space,  the  limb 
remaining  unchanged.  Thus,  if  the  full  lines 
in  Fig.  10  represent  the  positions  of  the 
microscopes  with  telescope  direct,  the  dotted 
lines  show  their  positions  with  telescope  re- 
Fig.  10.  verse.  In  this  lies  the  greatest  advantage 
of  three  microscopes  over  two,  since  with  the  latter,  in  revers- 
ing, the  microscopes  simply  change  places  with  each  other, 
without  reading  on  new  portions  of  the  limb. 

The  error  arising  from  want  of  level  of  the  horizontal  limb 
cannot  be  eliminated  by  the  method  of  observation,  but  with 
the  levels  which  accompany  a  good  instrument,  and  with  ordi- 
nary care,  it  will  usually  be  less  than  o.\" .  In  case,  however, 
of  a  signal  having  a  high  altitude  above  the  horizon,  the  error 
from  this  source  may  be  greater,  and  then  special  care  should 
be  taken  in  leveling.  For  an  expression  for  its  influence  in  any 
case,  see  Chauvenet's  Astronomy,  Vol.  II,  Art.  211. 

It  is  desirable  to  make  the  observations  under  various  condi- 
tions so  as  to  avoid  constant  errors.  See  Appendix  No.  4,  U.  S. 
C.  &  G.  Survey  Report  for  1903,  pp.  843-844,  869. 


ADJirST^IENT    OF    A    TRIANGULATION  —  ANGLES        185 

137.  We  shall  for  illustration  take  the  following  example, 
making  use  of  such  parts  of  it  from  time  to  time  as  may  belong 
to  the  subject  in  hand,  and  finally,  after  explaining  the  method 
of  forming  the  condition  equations,  solve  it  in  full. 

In  the  triangulation  of  Lake  Superior  executed  by  the  U.  S. 
Engineers  the  following  angles  were  measured  in  the  quadri- 
lateral N.  Base,  S.  Base,  Lester,  Oneota, 

LNO  =124°  09'  40.69"  weight    2 


SNL  = 

113°  39'  oS-o?" 

2 

/?>  / 

ONS  = 

122°  11'  15.61" 

14 

X/  / 

NSO  = 

23°  08'  05.26" 

23 

/      /'''    / 

LSN  = 

47°  31'  20.41" 

6 

/        /      / 

LSO  = 

70°  39'  24.60" 

7 

X    '/I     / 

SOJV=: 

34°  40'  39-66" 

3^           °' 

<i-^"    / 

NOL  = 

43°  46'  26.40" 

I 

\j^ 

OLS  = 

30°  53'  30-81" 

8 

s 

Fig.  II. 

se  angles  we  shall  denote 

by 

M^,  M^  .  . 

.  ,  M^  respectively. 

The  length  of  the  line  N.  Base —  S.  Base  (Minnesota  Point) 
is  6056.6  m.,  and  the  latitudes  of  the  four  stations  are  approxi- 
mately 

N.  Base,  46°  45'         Lester,    46°  52' 
S.  Base,  46°  43'         Oneota,  46°  45' 

138.  The  Local  Adjustment.  —  When  in  a  system  of  triang- 
ulation the  horizontal  angles  read  at  a  station  are  adjusted  for 
all  of  the  conditions  existing  among  them,  then  these  angles  are 
said  to  be  locally  adjusted. 

From  the  considerations  set  forth  in  Art.  133,  it  is  readily 
seen  that  at  a  station  only  two  kinds  of  conditions  are  possible  : 

(rt)  that  an  angle  can  be  formed  from  two  or  more  others, 
and 

(<^)  that  the  sum  of  the  angles  round  the  horizon  should  be 

equal  to  360''. 

The  second  of  these  is  included  in  the  fir.st,  and  the  method 
of  adjustment  may  be  stated  in  general  terms  as  follows: 


[86 


THE    ADJUSTMENT    OP    OBSERVATIONS 


An  inspection  of  the  figure  representing  the  angles  at  the 
station  will  show  how  all  of  the  measured  angles  can  be  ex- 
pressed in  terms  of  a  certain  number  of  them  which  are  inde- 
pendent of  one  another.  These  relations  will  give  rise  to 
condition  equations,  or  local  equations,  as  they  are  called,  which 
may  be  solved  as  in  Chapters  IV  or  V. 

Thus,  if  1/p  M ^,  .  .  .  M„  denote  the  single  measured  angles, 
and  f  J,  v.y,  .  .  .  v„  their  most  probable  corrections,  then  if  any 
of  the  angles  M/,,  Mj.  can  be  formed  from  others,  we  have,  by 
equating  the  measured  and  computed  values,  the  local  condition 
equations, 

M^   +   7',  =    J/i    +    Z'l  •+    ^^2    +    2^2   +     •    •    • 
Mj,  +   Vy,  =   M,    +    7-1   +  M.   +    7'2   +     •    •    . 


with 

where  A,  A'  •   • 


7'i  +  e'o  +   ■  ■ 

■  -  v„  =  /;,  suppose, 

Z'l  +  Z'2  +    •   • 

•  -  v^  =  4  suppose, 

■  4-  Pui'h'  +  A-7V'  +  ■  •  •  +  Pn^\?  =  a  minimum 
/„  denote  the  weights  of  the  angles. 

The  solution  may  be  in  general  best 
carried  out  by  the  method  of  correlates,  as 
in  Chapter  V. 

139.  The  following  special  cases  are  of 
frequent  occurrence  : 

(i)  At  a  station  O  the  n — i  single 
angles  A  OB,  BOC,  .  .  .  are  measured,  and 
also  the  sum  angle  AOL,  to  find  the  ad- 
justed values  of  the  separate  angles,  all  of  the 
measured  values  being  of  the  same  weight. 


The  condition  equation  is 
J/i  +  v^  +  J/o  +  ro  +  ■ 


+  J/;._i  +  t\. 


M„  +  V, 


or 


z\  +  7',  +  ■  •  •  +  z'„-i  -  ?■«  =  ^4  -  (^1  +  ^^2  + 

=  /  suppose, 
with  [7'^]  =  a  minimum. 


+  ^„-x) 


ADJUSTMENT   OF    A    TRIANGULATION —ANGLES        187 

The  solution  gives  (Art.  118  or  119), 

/ 

that  is,  tJic  correction  to  each  angle  is  -  of  the  excess  of  the  sum 

n 

ajigle  over  the  snin  of  the  single  angles,  and  the  sign  of  the  cor- 
rection to  the  sum  angle  is  opposite  to  that  of  the  single  angles. 

(2)  At  a  station  O  the  ;/  single  angles  ^4 (9^,  BOC,  .  .  .  LOA 
are  measured,  thus  closing  the  horizon,  to  find  the  adjusted 
values  of  the  angles. 

The  condition  equation  is 

z\  +  7'2  +  •  •  •  +  Vn  =  360°  -{M,  +  M^+  .  ■•  +  M„) 
=  /  suppose, 
with  [/z'^]  =  3.  minimum. 

The  solution  gives 

,        / 

[u] 


where  u.  =  -- ,  u^  =  ~- ,  •  ■  ■  and  [u]  =        I  • 

/i  A  L/J 

If  the  weights  are  equal,  then 

v,  =  v,=  ...  =7'„  =  -; 

that  is,  the  correction  to  each  angle  is    of  the  excess  of  360°  over 

the  sum  of  the  measured  angles. 

Ex.  I.  —  The  angles  at  Station  N.  Base  close  the  horizon;  required  to 
adjust  them. 

We  have  (Art.  137 j, 

yJ/,  +  7',  =  124°    09'    40.69"  +  7/,     weight  2 

^^2  +  7/2=113°      39'      05.07"  +  7A  "         2 

M^  +  "v^  =  122°  \\'  15.61^  +  7/3         "     14 

Sum =  360°  00'  01.37"  +  7/,  +  T..  +  7'a 

Theoretical  sum       .     .  =  360°  00^  00.00" 

.-.  Local  equation  is     .   =      o"  =  1-37  '  +  "^'i  +  ^'2  +  "3 


THE    ADJUSTMENT    OF    OBSERVATIONS 


Hence  (Ex.  2,  Art.  120),  7\  =—  -p— — \       _^    X  1.37 

=  —  0.64", 
V2  =  —  0.64", 
7/3  =  -  0.09", 
and  the  adjusted  angles  are, 

124°    09'    40.05" 

113°    39'    0443" 
122°     11'     15.52'' 


Check-sum  =  360°     00'     00.00" 
Ex.  2.  —  Precisely  as  in  the  preceding  we  may  deduce  at  Station  S.  Base 
the  local  equation, 

o  =  1.07"  +  Vi  +  v^-  v^, 
and  the  adjusted  angles 

23°    08'    05.13", 

47°    31'     19-91". 
70°    39'    25.04". 

140.  Number  of  Local  Equations  at  a  Station. — If  s  sta- 
tions are  sighted  at  from  a  station  that  is  occupied,  the  number 
of  angles  necessary  to  be  measured  to  determine  all  of  the 
angles  that  can  be  formed  at  the  station  occupied  is  j-  —  i.  If, 
therefore,  an  additional  angle  were  measured,  its  value  could  be 
determined  in  two  ways :  from  the  direct  measurement  and 
from  the  s  —  i  measures.  The  contradiction  in  these  two 
values  would  give  rise  to  a  local  (condition)  equation.  If,  there- 
fore, n  is  the  total  number  of  angles  measured  at  a  station,  the 
number    of    local    equations,    as    indicated  by    the    number    of 

superfluous  angles,  is 

n  —  s  +  I. 

141.  The  General  Adjustment. — With  a  single  measured 
base,  the  number  of  conditions  arising  from  the  geometrical  re- 
lations existing  among  the  different  parts  of  a  triangulation  net 
can  be  readily  estimated.  For  if  the  net  contains  s  stations, 
two  are  known,  being  the  end  points  of  the  base,  and  s  —  2  are 
to  be  found. 

Now,  two  angles  observed  at  the  end  points  of  the  base  will 
determine  a  third  point ;  two  more  observed  at  the  end  points 
of  a  line  joining  any  two  of  these  points  will  determine  a  fourth 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        1S9 

point,  and  so  on.  Hence,  to  determine  the  s  —  2  points,  2 
(s  —  2)  angles  are  necessary.  If,  therefore.  ;/  is  the  total  num- 
ber of  locally  independent  angles,  the  number  of  superfluous 
angles,  that  is,  the  number  of  conditions  to  be  satisfied,  is 

u  —  2  (s  —  2). 

Ex.  —  In  a  chain  of  triangles,  if  i-  is  the  number  of  stations,  show  that  tlie 
number  of  conditions  to  be  satisfied  is  j  —  2  ;  and  in  a  chain  of  quadrilat- 
erals, with  both  diagonals  drawn,  the  number  of  conditions  is  2  j  —  4. 

The  equations  arising  from  these  conditions  are  divided  into 
two  classes,  angle  equations  and  side  equations. 

142.  The  Angle  Equations.  — The  sum  of  the  angles  of  a 
triangle  drawn  on  a  plane  surface  is  equal  to  180°.  The  sum 
of  the  angles  of  a  spherical  triangle  exceeds  180°  by  the  spheri- 
cal excess  (e)  of  the  triangle,  which  latter  is  found  from  the 

relation 

area  of  triangle 

e  = y — I 77 > 

r^  sin  I 

r  being  the  radius  of  the  sphere. 

From  surveys  carried  on  during  the  past  two  centuries,  the 
earth  has  been  found  to  be  spheroidal  in  form,  and  its  dimen- 
sions have  been  determined  within  small  limits.  Now,  a 
spheroidal  triangle  of  moderate  size  may  be  computed  as  a 
spherical  triangle  on  a  tangent  sphere  whose  radius  is  "^TRJV, 
where  R,  N,  are  the  radii  of  curvature  of  the  meridian  and  of 
the  normal  section  to  the  meridian  respectively  at  the  point 
corresponding  to  the  mean  of  the  latitudes  <^  of  the  triangle 
vertices. 

Hence  we  may  wrap  our  triangulation   on    the   s|-)heroid  in 

question  by  conforming  it  to  the   spherical    excess    computed 

from  the  formula, 

a  b.  sin  C, 

e  (in  .seconds)  =  - — jyirf—- r, » 

^  '2  /t'yv  sm  I 

where  a^,  b^,  are  two  sides  and  C^  is  the  included  angle  of  the 
triangle. 


190  THE    ADJUSTMENT    OF    OBSERVATIONS 

For  convenience  of  computation  we  may  write, 

e  ^=  inaj\  sin  Cj, 

when  log  m  may  be  tabulated  for  the  argument  <^.  The  follow- 
ing table  is  computed  with  Clarke's  values  of  the  elements  of 
the  terrestrial  spheroid  of  1866  corresponding  to  latitudes  from 
10°  to  70^.     The  meter  is  the  unit  of  length  to  be  used. 

Table  of  log  in. 


Lati- 

Log ;«. 

Lati- 

Log m. 

Lati- 

Log ;«. 

La 

t- 

Log  m. 

tude. 

tude. 

tude. 

tuije. 

0    / 

0 

/ 

0 

/ 

0 

, 

18   00 

T.40639 

33 

00 

T.40520 

48 

00 

T.40369 

63 

00 

r.40227 

18   30 

636 

33 

30 

516 

48 

30 

364 

63 

30 

223 

19   00 

632 

34 

00 

511 

49 

00 

359 

64 

00 

219 

19   30 

629 

34 

30 

506 

49 

30 

354 

64 

30 

215 

20   00 

626 

35 

00 

501 

50 

00 

349 

65 

00 

210 

20   30 

623 

35 

30 

496 

50 

30 

344 

65 

30 

207 

2  1   00 

619 

36 

00 

491 

51 

00 

339 

66 

00 

203 

21   30 

616 

36 

30 

486 

51 

30 

334 

66 

30 

199 

22   00 

612 

37 

00 

4S2 

52 

00 

329 

67 

00 

195 

22   30 

608 

37 

30 

477 

52 

30 

324 

67 

30 

192 

23   00 

605 

38 

00 

472 

53 

00 

319 

68 

00 

188 

23   30 

601 

38 

30 

467 

53 

30 

314 

68 

30 

185 

24   00 

597 

39 

00 

462 

54 

00 

309 

69 

00 

181 

24   30 

594 

39 

30 

457 

54 

30 

304 

69 

30 

178 

25   00 

590 

40 

00 

452 

55 

00 

299 

70 

00 

174 

25   30 

586 

40 

30 

446 

55 

30 

295 

70 

30 

171 

26   00 

5S2 

41 

00 

441 

56 

00 

290 

71 

00 

168 

26   30 

578 

41 

30 

436 

56 

30 

285 

71 

30 

164 

27   00 

573 

42 

00 

431 

57 

00 

280 

72 

00 

T.40161 

27   30 

569 

42 

30 

426 

SI 

30 

276 

• 

28   00 

565 

43 

00 

421 

58 

00 

271 

28   30 

560 

43 

30 

4.6 

58 

30 

266 

29  CO 

556 

44 

00 

411 

59 

00 

262 

29   30 

552 

44 

30 

406 

59 

30 

257 

30   00 

548 

45 

00 

400 

60 

00 

253 

30   30 

544 

45 

30 

395 

60 

30 

249 

31   00 

539 

46 

00 

390 

61 

00 

244 

31   30 

534 

4^' 

30 

385 

61 

30 

240 

32   00 

530 

47 

00 

380 

62 

00 

•235 

32   30 

1.40525 

47 

30 

1-40375 

62 

30 

1-40231 

ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        19k 

To  find  a^,  b^,  cf),  a  preliminary  geodetic  computation  must  first 
be  made  of  the  triangulation  to  be  adjusted,  starting  from  a 
base  or  from  a  known  side.  The  values  found  from  using  the 
unadjusted  angles  will  be  close  enough  for  this  purpose.  An 
error  of  less  than  3'  in  the  latitude  will  not  under  any  circum- 
stances produce  an  error  of  overo.ooi"  in  the  computed  spheri- 
cal excess,  and  in  general  therefore  the  latitudes  may  be  taken 
from  a  map  or  sketch  of  the  triangulation. 

143.  A  useful  check  of  the  excess  results  from  the  principle 
that  the  sums  of  the  excesses  of  triangles  that  cover  the  same 
area  should  be  equal.  In  our  example  the  spherical  excesses 
of  the  triangles  OXS,  LSO  will  be  found  to  be  0.05"  and  0.37" 
respectively. 

In  each  single  triangle,  then,  the  condition  required  to  wrap 
it  on  the  spheroid,  that  is,  that  the  sum  of  the  three  measured 
angles  shall  be  equal  to  180°,  together  with  the  spherical  excess, 
gives  a  condition  equation.*  This  is  called  an  a;/^/e  equation, 
or  by  some  a  triangle  equation. 

Ex.  —  In  the  triangle  N.  Base,  S.  Base,  Oneota,  if  v.,,,  7'.,,  ?'„  denote  the 
corrections  to  the  three  angles,  we  have  for  the  most  probable  values, 

ONS  =  122°  II'  15.61"  +  7/3 
NSO  =  23°  08'  05.26"  +  v^ 
SON  =    34°    40'    39.66"  +  v^ 


Sum     .     .     .     .    =  180°    00'    00.53"  +  "^^3  +  'Z'4  +  th 
Theoretical  sum  =  180°    00'    00.05"  =  'S°°  +  ^ 

and  the  angle  equation  is  formed  by  equating  these  sums.     The  result  is, 
T's  +  Vi  +  v-i  +  0.48"  =  o. 
Similarly,  from  the  triangle  Lester,  S.  Base,  Oneota,  the  angle  equation  is, 

V^  +  V^  +  7/3  +  7/9  +   1. 10"  =-  O. 

144.  Number  of  Angle  Equations  in  a  Net.  —  It  is  to  be 
expected  that  in  a  triangtilation  net  some  of  the  lines  will   bo 

*  We  confine  ourselves  throughout  to  triangles  to  which  Lcgendre's  theo- 
rem is  applicable.  For  very  large  triangles  other  formulas  for  spherical 
excess  must  be  used  if  great  accuracy  is  required.  See  The  Transcoiitntnital 
Triangulation  of  C.  &'  G.  Survey,  pp.  51-54. 


192  THE    ADJUSTMENT    OF    OBSERVATIONS 

sighted  over  in  both  directions,  and  some  only  in  one  direction. 
If  these  latter  lines  are  omitted,  the  number  of  angle  equations 
will  remain  unaltered.  Thus  in  our  Lake  Superior  quadrilateral 
(Fig.  1 1)  the  line  NL  has  been  sighted  over  from  N,  but  not  from 
Z,  so  that  we  have  only  two  angle  equations  :  namely,  those  re- 
sulting from  the  triangles  ONS,  OLS,  just  as  if  the  figure  had 
been  of  the  form  of  Fig.  1 3,  in  which  the  line  NL  is  omitted. 

Let  s  be  the  total  number  of  stations  in  a  figure  or  series  of 
figures  regardless  of  whether  the  stations  are  occupied  or  not, 
j-„  the  number  of  unoccupied  stations,  /  the  total  number  of  lines 
in  the  figure,  and  /^  the  number  of  lines  which  are  observed 
over  in  one  direction  only.  The  number  of  angle  equations 
will  be  /  —  /j  —  J-  +  ^„  +  /. 

This  may  be  proved  by  plotting  the  figure  in  detail,  adding  at 

each  step  of  the  process  one  new 
point  and  all  observed  lines  connect- 
ing that  point  with  points  already 
shown  on  the  figure. 

The  formula  is  true  in  any  case, 
because,  as  the  figure  is  drawn  point 
by  point  and  line  by  line  as  indi- 
cated, it  holds  for  the  simplest  pos- 
sible figure  the  triangle ;  for  the 
first  two  lines  to  any  new  occupied 
point,  /  —  J-  is  increased  by  one  and 
the  value  of  no  other  symbol  is  changed  in  the  formula  and  one 
new  angle  equation  appears  ;  for  each  new  complete  line  to  a  new 
occupied  point  after  the  first  two,  /  is  increased  by  one,  and  one 
new  angle,  equation  appears  ;  for  each  new  line  observed  in  one 
direction  only  to  a  new  point  after  the  first  two  lines  to  it  are 
drawn,  I  —  l^  remains  unchanged,  and  no  new  angle  equation 
appears ;  the  addition  of  an  unoccupied  station  with  any  number 
of  lines  to  it  which  are  necessarily  observed  in  one  direction 
only  does  not  change  the  value  oi  I  —  l^  —  s  ■\-  s^  -{-  I,  and  no 
new  angle  equation  is  introduced. 


ADJUSTMENT    OF    A    TRIANGULATION —ANGLES       193 


It  may  seem  that  there  are  more  angle  equations  than  have 
been  indicated,  but  it  will  be  found  in  every  such  case  that  the 
supposed  additional  equation  may  be  derived  algebraically  from 
those  already  used,  and  is  therefore  not  a  new  independent 
equation. 

Ex. —  In  the  quadrilateral  A  BCD,  in  which  all  of  the  8  angles  are  meas- 
ured, show  that  there  are  three  independent  angle 

equations,  and  that  these  may  be  found  from  the  fol-       a    -  B 

lowing  8  sets  of  figures : 

ABD,  ABC,  A  CD;  ABD,  ABC,  ABCDj 
ABD,  ACD,  ABCD; 

BDA,  BCD,  BCAj  BCA.  BCD,  BCDA; 
CDB,    CAB,  CD  A;  CDB,  CD  A,  CDBA; 
DAB,  DBC,  DAC. 


Fig.  14. 


145.  The  Side  Equations.  —  In  a  single  triangle,  or  in  a 
simple  chain  of  triangles,  the  length  of  any  assigned  side  can  be 
computed  from  a  given  side  in  but  one  way.  When  the 
triangles  are  interlaced,  this  is  not  the  case. 

Thus  in  Fig.  13  any  side  can  be  computed  from  NS  in  but 
one  way.  The  only  condition  equations  apart  from  the  local 
,L2  equations  would  be  the  two  angle 
equations.  But  in  Fig.  1 1,  in  which 
the  line  NL  is  sighted  over  from  N, 
we  have  the  further  condition  that 
the  lines  OL,  NL,  SL  intersect  in 
the  same  point,  L.  The  figure 
plotted  from  the  measured  values 
would  be  of  the  form  of  Fig.  15. 

To  express  in  the  form  of  an  equa- 
tion   the    condition    that    the    three 
F's-'S-  points  /.,,  L.,,  jC.j  must  coincide,  we 

proceed  as  follows  :  Starting  from  the  base  NS,  we  may  com- 
pute 5Z,  directly  from  the  triangle  SNL^,  and  SL.^  from  the 
triangles  SON,  SOL,.     This  gives 

side  SN  _  sin  SL^N 
side  SL^       sin  SN/.^ ' 


194  THE    ADJUSTMENT    OF    OBSERVATIONS 

side  SJV  _  sin  SOJV  sin  SZ^O . 
side  SZs       sin  SJVO  sin  SOL^ 

but  SL^  must  be  equal  to  SL^. 
Hence  the  condition  equation  is 

sin  SLN  sin  SNO   sin  SOL  _ 
sin  SNL  sin  SON  sin^Z6>  "  ^' 

which  is  called  a  jzV/i?  equation  or  j/;/^  equation. 
The  side  equation, 

sin  ^ZiV^  sin  SOL   sin  ^A^(9  _ 
sin  SNL   sin  6'Z6>  sm^SWV^  ~  ^' 

gives  the  identical  relation, 

side  SN  side  .S'Z    side  SO   _ 
side  ,SZ    side  SO   side  »SA^ 

Hence,  in  forming  a  side  equation  we  may  proceed  mechani- 
cally in  this  way.     Write  down  the  scheme 

SN  SL^  S0_ 
SL  SO  SN~  ^' 

the  numerator  and  denominator  each  being  formed  by  the  lines 
radiating  from  the  point  5  in  order  of  azimuth,  and  the  first 
denominator  being  the  second  numerator.  The  side  equation 
results  from  replacing  the  sides  by  the  sines  of  the  angles 
opposite  to  them. 

The  point  5  is  called  the  J^o/e  of  the  quadrilateral  for  this 
equation. 

It  should  be  noted  that  side  equations  formed  from  a  pole  as 
starting-point  are  the  most  convenient  to  be  used  of  the  many 
that  are  possible.  Not  necessarily  do  all  of  the  lines  radiate 
from  a  point  or  pole  that  enter  a  side  equation. 

The  side  equations  thus  formed  by  the  use  of  lines  radiating 
from  a  selected  pole  are  not  the  only  ones  possible.  They  are 
the  convenient  ones.  As  an  example  of  a  side  equation  not 
formed  with  a  pole,  take  the  following  from  Fig.  23,  supposing 
Zj,  Zg,  Z3,  to  form  one  point  L.     The  equation  is 


ADJUSTMENT    OF    A    TRIANGULATION —  ANGLES       195 

sin  ONS  sin  NLS  sin  NOL  sin  LSO  ^ 
sin  SON  sin  LSN  sin  ZA^C>  sin  6>Z^  ~  '' 

This  equation  expressed  in  the  form  of  ratios  of  sides  is 

SO  SNNL  0L_ 
'SNNL  'OLSd~^' 

It  is  evident  that  the  sides  involved  do  not  il  radiate  from 
one  point.  Such  side  equations  as  these  need  not  be  used,  as  a 
sufficient  number  of  the  more  conveniently  formed  side  equa- 
tions of  the  kind  which  involve  a  pole  may  always  be  secured. 

Ex.  —  In  the  figure  ABCD^D^A,  the  three  angle  equations, 

D,AB  +  ABD^  +  BD,A  =  180°  +  e^, 

ABC    +  BCA    +  CAB   =  180°  +  e,, 

BCD^  +  CZ>35  +  D^BC  =  180°  +  63, 
given  by  the  triangles  Z>,yi 5,  ABC,  BCD^,  may- 
be satisfied,  and  yet  the  figure  not  be  a  perfect 
quadrilateral.  Show  by  equating  the  values 
of  BD^  and  BD^  that  the  further  condition 
necessary  is 

sin  ABD   sin  BCA    sin  CDB  _ 

sin  BDA    sin  CAB   sin  BCD  ~    '  Fig.  16. 

146.  Position  of  Pole.  —  It  is  easily  seen  that  in  formin*^- 
the  side  equation  any  vertex  may  be  taken  as  pole.  For  plot- 
ting the  figure  from  the  angles  of  the  triangles  ONS,  OLS,  the 
side  equation  with  pole  at  5  means  that  the  points  L^  and  L^ 
must  coincide.  The  side  equation  with  pole  at  N  means  that 
Zj,  L^  coincide,  and  with  pole  at  O  that  L.,,  L.^  coincide.  If 
any  one  of  these  conditions  is  satisfied,  the  others  are  also  satis- 
fied, as  each  amounts  to  the  same  condition  that  L  is  not  three 
points,  but  one  point. 

Similar  reasoning  will  show  that  by  plotting  the  figure  from 
LONS,  ONS,  the  side  equations  formed  by  taking  the  poles  at 
N,  L,  S,  mean  that  O  is  not  three  points  but  one  point,  and  so 
on.  Hence  the  side  equation  formed  from  any  vertex  as  pole  in 
connection  with  the  angle  equations  fixes  each  point  of  the 
figure  definitely  and  removes  all  contradictions  from  it. 


196  THE    ADJUSTMENT    OF    OBSERVATIONS 

It  will  be  noticed  that  the  reasoning  is  in  no  way  affected 
by  the  Une  NL  being  sighted  over  in  only  one  direction, 

Ex.  I.  —  In  a  quadrilateral  ABCD,  in  which  all  of  the  8  angles  are 
measured,  show  that  of  the  15  side  equations  that  may  be  formed,  not  all  of 
which  are  of  the  polar  kind,  7  only  are  different  in  form,  and  that  by  taking 
the  angle  equations  into  account,  all  of  them  may  be  reduced  to  a  single  form. 

Also  show  that  there  are  56  ways  of  expressing  the  3  angle  and  i  side 
equations  necessary  to  determine  the  quadrilateral. 

Ex.  2.  —  Examine  the  truth  of  the  following  statement.  In  a  quadri- 
lateral an  angle  equation  may  be  replaced  by  a  side  equation,  so  that  the 
quadrilateral  may  be  determined  by  3  angle  equations  and  i  side  equation, 
2  angle  equations  and  2  side  equations,  one  angle  equation  and  3  side  equa- 
tions, the  number  of  conditions  remaining  four,  and  the  four  not  being  all 
of  one  kind. 

If  the  triangulation  net,  instead  of  involving  quadrilaterals 
only,  involves   central  polygons,  such   that,  in   computing    the 

lengths  of  the  sides,  we  can  pass 
from  one  side  to  any  other  through 
a  chain  of  triangles,  the  same  pro- 
cess is  followed  in  forming  the 
side  equations  as  in  a  quadri- 
lateral. 

Thus,  in  the  figure  which  repre- 
sents part  of  the  triangulation  of 
Lake  Erie  west  of  Buffalo  Base, 
there  are  side  equations  from 

The  quadrilaterals  CDHG,  GHFA, 
The  pentagons  GABCH,  HGDEF. 

The  scheme  for  the  pentagonal  side  equation  GABCH,  for  ex- 
ample, would  be  just  as  in  the  case  of  a  quadrilateral,  taking 
G  as  pole, 


GA    GB   GC  GH  ^ 
GB  GC  GH    GA       ^' 


and  the  side  equation. 


ADJUSTMENT    OF    A    TRIAXGULATION  —  ANGLES       197 
sin  GBA    sin  GCB   sin  GHC  sin  GAH 


sin  GAB   sin  GBC   sin  GCH  sin  G'/i'l-/ 

147.  Reduction  to  the  Linear  Form.  —  Thus  far  we  have 
considered  the  side  equations  in  their  rigorous  form.  But  in 
order  to  carry  through  the  solution  by  combining  them  with 
the  other  condition  equations,  they  must  be  reduced  to  the  linear 
form.     We  proceed  to  show  how  this  may  be  done. 

Let  the  side  equation  be 

sin  J^   sin  f^ 

sin   V^   sin   V^'  '  "  =  ^'  ^'^ 

where  V^,  l\,>  •  •  •  denote  the  most  probable  values  of  the 
angles.  Let  Jll^,  M^,  .  .  .  denote  the  measured  values,  and 
v^,  v^,  .  .  .  the  most  probable  corrections  to  these  values ; 
then  the  equation  may  be  written, 

sin  {M^  +  v^   sin  (J/3  +  ^'s)         ^  ^  ,  . 

sin  (J/,  +  v^  sin  (J/^  +  v^  •  ■  •       i.  (.2; 

Taking  the  log  of  each  side  of  this  equation,  and  expanding  by 
Taylor's  theorem,  we  have,  retaining  the  first  powers  of  the 
corrections  only, 

log  sin  M^  +  j^  (log  sin  M^  i\ 

-  j  log  sin  M.,  +  j^  (log  sin  J/,)  v.,  f  +  .  .  .  =  o,    (3) 

which  may  be  written  in  two  forms  for  computation  : 

First,  if  the  corrections  to  the  angles  are  expressed  in  sec- 
onds, we  may  put 

-^  (log  sin  M.)  =  8'. 

where  V  is  the  tabular  difference  for  \"  for  the  angle  M^  in  a 
table  of  log  sines.     Then  we  have, 

8'z/j  —  8" 7^2  +  •  •  •  +  log  sin  M^  —  log  sin  M.,-\-  ■  ■  •  =  o\ 
that  is, 

[8^]  =  ^.  (4) 

where  /  is  a  known  quantity. 


198  THE    ADJUSTMENT    OP    OBSERVATIONS 

nay  replace 
(log  sin  J/j)  by  mod  sin  i"  cot  M^, 


Secondly,  we  may  replace 


where   mod   denotes   the  modulus   of   the   common    system   of 
logarithms.     Eq.  3  may  then  be  arranged, 
cot  M^t\  —  cot  M.{i)^  +  •  •  • 

=  —. \r- r,  (log  sin  J/.  -  log  sin  J/j  -f  •  •  •),  (s) 

10'  mod  sm  i 

if  the  seventh  place  of  decimals  is  chosen  as  the  unit. 

The  first  of  these  two  forms  is  preferred  in  the  computing 
work  by  the  computers  of  the  Coast  and  Geodetic  Survey. 

148.  Check  Computation.  —  The  side  equation  deduced 
from  spherical  triangles  must  also  follow  from  the  correspond- 
ing plane  triangles,  the  angles  of  each  spherical  triangle  being 
transformed  according  to  Legendre's  theorem  ;  that  is,  for  ex- 
ample, we  should  obtain  the  same  constant  term  /  by  reducing 
to  the  linear  form  the  equation, 

sin  SLN  sin  SOL    sin  SNO    _ 
sin  SNL    sin  SLO   sin  SON  ~  ^  ' 

or  the  equation. 


sin 


[sLN-  "i)   sin  {sol  -  -)   sin  (sNO  -  "?) 


sm 


[sNL  -  "i)   sin  {sLO  -  -)    sin  isON -  "") 


where  e,,  e.,,  e^,  denote  the  spherical  excesses  of  the  triangles 
SNL,  SOL,  and  SON,  respectively. 

It  is,  in  general,  simpler  to  use  the  spherical  angles  than  the 
plane  angles.  It  affords  a  check  on  the  accuracy  of  the  numeri- 
cal work  to  compute  the  side  equation  with  both  the  spherical 
and  the  plane  angles.  It  is  hardly  worth  while,  however,  to 
spend  the  time  required  for  this  check,  as  it  will  take  as  long  to 
apply  the  check  as  to  have  a  duplicate  computation  made. 

Ex. —  The  quadrilateral  N.  Base,  S.  Base,  Oneota,  Lester  (Fig.  19). 
Take  the  pole  at  Lester. 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES         199 

,.r    u         ,1  L^    LN  LO 

We  have  the  scheme  y-^   j-^    -j-^  ^  ^' 

from  which  we  write  down  the  side  equation, 

sin  LNS   sin  LON  sin  LSO  ^ 

em  LSjV  sin  LA^O   sin  jL06'      ^  ' 

sin  (.4/^  +  7/0)   sin  (Afs  +  -z/g)  sin  (M^,  +  v^  ^  ^ 

that  IS,       ^.^  ^^j^^  _l_  ^^^   gjj^  ^^^^  _l_  ^^^   gjj^  ^^^  +  j/^  +  ^^  +  ^^  " 

^/r^/  Form  of  Reductiott. 
log  sin  (113°  39'  05.07"  +  7/2)  =  9.9618970  -    9.2  2/, 

log  sin  (  43°  46'  26.40"  +  v^)  =  9-8399903  +    2.0  z/g 

log  sin  (  70°  39'  24.60"  +  ■z/g)  =  9-9747657  +    74  ■z'a 

530 
log  sin  (  47°  31'  20.41"  +  7/5)  =  9.8677860  +  19.3  T/5 

log  sin  (124°  09'  40.69"  +  v^)  =  9.9177470  -  14.3  Vi 

log  sin  (  78°  27'  06.06"  +  7/7  +  T/g)  =  9.991 1^80  +    4.3  (v.,  +  z/,) 

510 
Hence  the  side  equation  in  the  linear  form  is 

14.3  v^  -  9.2  V2  -  19.3  v^  +  7.4  7/a  -  4-3  ^7  +  17.7  7/g  +  20  =  o, 
the  unit  being  the  seventh  place  of  decimals. 

Check  off  the  constant  term  by  computing  the  log  sines  after  deducting 
from  each  angle  ^  of  the  spherical  excess  of  the  triangle  to  which  it  belongs. 

Angle.  Log  Sin.  Angle.  Log  Sin. 

113°  39'  05.00"  9.9618970  47°  31'  20.34"  9.8677858 

43°  46'  26.36"  9.8399903  124°  09'  40.65"  9.91 7747 1 

70°  39'  24.48"  99747656  78°  27'  05.02"  9.991 1 180 

529  509 

4-  20 

agreeing  closely  with  the  value  found  from  the  spherical  angles. 
Second  Form  of  Reduction. 

Log  Sin.  Log  Sin. 

9.9618970  —  0.438  7/2  9.8677860  +  0.916  Vf, 

9.8399903  +  1.044  7/g  9.9177470  -  0.679  7/, 

9.9747659  +  0.351  V  9.9911180  +  0.204  (7^7  +  ^3) 
530 

51° 
20    log  1. 30103 

— \—. — r,  log  8.67664 

10'  mod  sm  i  

9.97767      0.955 
and  the  side  equation  is 

0.679  7/,  -  0.438  V.^  -  0.916  7^5  4-  0.351  Vn  -  0.204  V^  +  0.840  7/,  +  0.95   =  O. 

This  result  may  be  checked  in  the  same  way  as  in  the  first  form. 

In  reducing  a  side  equation  to  the  linear  form,  the  coefficients  of  the  cor- 
rections should  be  carried  out  to  one  place  of  decimals  farther  than  the 


ioo  THE    ADJUSTMENT    OF    OBSERVATIONS 

absolute  term.  This  for  a  short  computation  would  be  unnecessary,  but  in 
the  reduction  of  an  extensive  triangulation  net  it  is  rendered  necessary  by 
the  accumulation  of  errors  from  the  dropping  of  the  last  figures  in  products 
and  quotients. 

It  will  be  noticed  that  in  the  preceding  example  logarithmic 
sines  have  been  carried  to  seven  decimal  places  only.  This  is 
sufficient  for  the  most  accurate  primary  triangulation.  An  error 
of  one  in  the  seventh  place  of  decimals  corresponds  to  an  error  of 
less  than  i  part  in  4,000,000.  In  the  most  accurate  primary  tri- 
angulation, discrepancies  of  more  than  ten  times  the  amount  occur 
in  at  least  half  of  the  figures.  The  uniform  present  practice  of 
the  Coast  and  Geodetic  Survey  is  to  use  7-place  logarithms.  In 
the  past,  eight  or  more  places  have  been  frequently  used. 

149.  We  have  seen  that  the  coefificients  of  the  corrections  in 
a  side  equation  are  given  by  the  differences  for  i"  of  the  log 
sines  of  the  angles,  or  by  the  cotangents  of  the  angles  that  enter. 
There  will  be  less  liability  to  mistakes  on  account  of  misplaced 
decimal  points,  and  less  difficulty  arising  from  omitted  decimal 
places  in  the  solution,  and  especially  in  connection  with  a  check 
column,  if  the  coefficients  throughout  the  condition  equations 
are  of  the  same  order  of  magnitude.  Since  the  coefficients  of 
the  corrections  in  the  angle  equations  are  +  i  or  —  i,  it  fol- 
lows that  it  would  be  most  convenient  to  put  the  side  equations 
on  the  same  footing  as  the  angle  equations.  To  do  this  we  may 
divide  the  side  equation  by  such  a  number  as  will  make  the 
average  value  of  the  coefficients  equal  to  unity.  This,  for  angles 
ordinarily  met  with  in  triangulation,  would  be  effected  by  taking 
the  sixth  place  of  decimals  as  the  unit  in  the  side  equation. 
Thus  in  our  example,  dividing  by  10,  which  is  approximately  the 
mean  of  the  coefficients,  and  which  amounts  to  the  same  thing 
as  expressing  the  log  differences  in  units  of  the  sixth  place  of 
decimals,  the  equation  may  be  written 

1-43  ^1  —  0-92  2^2  —  1-93  ^'5  4-  0.747^6  —  0-43  ^'7  +  1-77  ^'s  +  2.00  =  o. 

It  would  have  been  equally  correct  to  multiply  each  of  the 

angle  equations  by   10,  and  so  have  put   them  on  the  same 


ADJUSTMENT    OF    A    TRIANGULATION —  ANGLES       201 

footing  as  the  side  equations.  Dividing  the  side  equations  is, 
however,  to  be  preferred,  as  the  coefificients  are  made  smaller 
throughout,  and  the  formation  and  solution  of  the  normal 
equations  are  consequently  easier. 

A  striking  difference  between  condition  equations  and  obser- 
vation equations  is  here  brought  out.  As  a  condition  equation 
expresses  a  rigorous  relation  among  the  observed  quantities  al- 
together independent  of  observation,  it  may  be  multiplied  or 
divided  by  any  number  without  affecting  that  relation  ;  with  an 
observation  equation,  on  the  other  hand,  the  effect  would  be  to 
increase  or  diminish  its  weight.     (Compare  Art.  48.) 

150.  Position  of  Pole.  —  In  a  quadrilateral,  taking  any  of  the 
vertices  as  pole,  the  conclusion  was  reached  in  Art.  145  that  any 
one  of  the  resulting  forms  of  side  equation  was  as  good  as  any 
other  in  satisfying  the  conditions  imposed.  But  when  a  side 
equation  is  reduced  to  the  linear  form  and  is  no  longer  rigorous, 
the  question  deserves  further  notice. 

Two  points  are  to  be  considered  —  precision  of  results  and 
ease  of  computation.  As  regards  the  first,  since  the  differences 
in  a  table  of  log  sines  are  more  sharply  defined  for  small  angles, 
and  these  differences  are  the  coefficients  of  the  unknowns  in  the 
side  equation,  it  follows  that  in  general  that  vertex  should  be 
chosen  which  allows  the  introduction  of  the  acutest  angles  into 
the  side  equation. 

Labor  of  computation  will  be  saved  by  choosing  the  pole  so 
that  as  few  sine  terms  as  possible  enter.  Thus  by  choosing  the 
pole  at  O,  the  intersection  of  the  diagonals  (Fig.  22),  the  side 
equation  would  contain  8  terms,  wherea.s,  if  taken  at  any  of  the 
vertices,  only  6  terms  would  enter.  Also,  other  things  being 
equal,  we  should  choose  that  pole  which  introduces  the  smallest 
number  of  unknowns  into  the  equation,  for  then  the  normal 
equations  would  be  more  easily  formed. 

If  the  approximate  form  of  solution  in  Art.  131  is  employed, 
it  is  advantageous  to  choose  the  pole  at  the  intersection,  O,  of 
the  diagonals,  as  will  be  seen  in  the  sequel. 


202  THE    ADJUSTMENT    OP    OBSERVATIONS 

151.  Number  of  Side  Equations  in  a  Net.  —  A  line  being 
taken  as  a  base,  its  extremities  are  known.  To  fix  a  third  point, 
we  must  know  the  other  two  sides  of  the  triangle  of  which  this 
point  is  to  be  the  vertex.  Hence  if  we  have  a  net  of  triangles 
connecting  s  stations,  two  of  the  stations  being  the  ends  of  the 
base,  we  must  have,  in  order  to  plot  the  figure,  2  {s  —  2)  lines 
besides  the  base ;  that  is,  2  s  —  2,  lines  in  all. 

Starting  from  the  base,  each  line  in  this  figure  can  be  com- 
puted in  but  one  way,  but  any  additional  line,  whether  observed 
over  in  one  or  both  directions,  can  be  computed  in  two  ways, 
and  therefore  gives  rise  to  a  side  equation.  If,  then,  the  total 
number  of  lines  in  the  figure  is  /,  the  number  of  side  equations, 
as  indicated  by  the  number  of  superfluous  lines,  is 

/—  2  i-+  3. 

152.  Check  of  the  Total  Number  of  Conditions.  —  The  no- 
tation already  given  may  be  summarized  as  follows :  In  any  figure 
or  series  of  figures,  /  is  the  total  number  of  lines,  /^  the  number 
of  lines  observed  over  in  one  direction  only,  s  the  total  number 
of  stations,  s,^  the  number  of  unoccupied  stations,  and  ;/  the 
number  of  angles  measured  which  are  independent  in  so  far  as 
local  conditions  are  concerned. 

At  each  station  occupied,  the  number  of  locally  independent 
angles  is  one  less  than  the  number  of  lines  observed  from  that 
station,  hence 

«  =  2  /  —  /j  —  i-  -f  j„.  (i) 

From  Art.  144  the  number  of  angle  equations  in  the  figure  is 

/  -  /,  -  .y  +  x„  +  /.  (2) 

From  Art.  1 5  i  the  number  of  side  equations  in  the  net  is 

/  -  2  J  +  3.  (3) 

More  accurately,  this  is  the  number  of  side  equations  which  are 
independent  of  each  other  and  of  the  angle  equations. 

Adding  (2)  and  (3),  the  total  number  of  angle  and  side  equa- 
tions in  the  net  is 

2  /  -  /^  -  3  J  +  J„  +  4- 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       203 

Combining  these  with  (i),  it  becomes  ;/  —  2  x  4-  4,  as  was 
proved  in  Art.  141  to  be  the  total  number  of  conditions  in  a 
figure  developed  from  a  single  base. 

The  condition  equations  referring  to  lengths,  azimuth,  latitude 
and  longitude,  which  arise  when  there  is  more  than  one  line  in 
a  figure  fixed  by  previous  adjustment,  will  be  treated  later. 

153.  Manner  of  Selecting  the  Angle  and  Side  Condition 
Equations.  — In  the  selection  of  side  and  angle  equations  in  a 
triangulation  net,  four  dangers  must  be  guarded  against.  First, 
that  some  necessary  condition  equation  may  be  omitted ;  second, 
that  some  unnecessary  condition  equation  may  be  introduced ; 
third,  that  a  condition  equation  which  is  chosen  may  not  be  in- 
dependent of  those  already  selected ;  and  fourth,  that  the  con- 
dition equations  used  may  be  so  selected  from  the  many  available 
that  the  solution  of  the  normal  equations  will  be  an  unstable 
one ;  that  is,  a  solution  in  which  the  effect  of  omitted  decimal 
places  on  the  derived  values  of  the  required  unknowns  is  large, 
and  in  which  it  is  therefore  necessary  to  carry  a  large  number 
of  decimal  places  in  the  solution  to  secure  the  unknowns  with 
certainty  to  a  small  number  of  decimal  places. 

A  good  method  of  avoiding  the  first,  second,  and  third  of 
these  dangers  is  to  start  from  some  line  as  base  and  plot  the 
figure  point  by  point.  As  each  point  is  added,  draw  all  observed 
lines  connecting  it  with  points  previously  located,  and  express 
the  conditions  arising  from  the  new  lines.  For  each  new  point, 
the  number  of  new  angle  equations  is  one  less  than  the  num- 
ber of  lines  observed  in  both  directions  connecting  it  with  pre- 
vious points,  and  the  number  of  new  side  equations  is  two  less 
than  the  number  of  lines  connecting  it  with  previous  points, 
regardless  of  whether  these  lines  are  observed  in  both  directions 
or  only  in  one  direction. 

For  example,  let  Fig.  18  represent  a  triangulation  net,  plotted 
in  detail  as  follows:  First  draw  the  line  connecting  the  points 
Tobacco  Row  and  Spear.     Add  the  new  point  Long,  and  con- 


204 


THE    ADJUSTMENT    OF    OBSERVATIONS 


TOBACCO  ROW 


nect  with  Tobacco  Row  and  Spear.  This  furnishes  one  angle 
equation  from  the  triangle  Long-Tobacco  Row-Spear.  Next 
add  the  new  point  Smith,  and  draw  lines  from  it  to  Tobacco 

Row,  Spear, and  Long. 
R  This  furnishes  two 
new  angle  equations 
corresponding  to  the 
two  new  triangles, 
and  one  side  equation. 
Complete  the  figure 
by  adding  the  new 
point  Flat  Top,  and 
draw  four  lines  to 
Tobacco  Row,  Spear, 
^"'^"  Long,      and     Smith. 

This  introduces  three 
new  angle  equations  and  two  new  side  equations,  making  the 
total  number  of  angle  equations  six,  and  of  side  equations  three. 
These  numbers  may  be  checked  by  the  formulas  in  sec.  i  54. 

If  in  this  triangulation  net  the  line  Spear-Smith  had  been 
observed  in  but  one  direction.  Spear  to  Smith,  and  Flat  Top 
had  been  an  unoccupied  station,  the  drawing  of  the  figure  in  the 
same  three  steps  would  have  indicated  respectively  one  angle 
equation,  one  angle  equation  and  one  side  equation,  and  two  side 
equations.  The  total  number  of  side  equations  would  have  been 
three  as  before,  and  of  angle  equations  two.  These  numbers 
may  be  checked  by  the  formulas  in  sec.  154. 

154.  If  the  above  process  of  selecting  the  condition  equations 
is  followed  strictly,  there  will  be  little  danger  of  choosing  mutu- 
ally dependent  condition  equations.  If,  however,  such  mutually 
dependent  equations  have  been  chosen,  it  will  become  evident 
in  the  course  of  the  solution  by  the  appearance  there  of  two 
equations  which  are  identical.  In  this  case  one  of  the  corre- 
lates becomes  indeterminate.  The  danger  of  selecting  such 
condition  equations  that  the  solution  will  be  somewhat  unstable 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        205 

is  a  much  more  difficult  one  to  avoid.     In  such  cases  the  skill 
of  the  expert  computer  gives  him  a  decided  advantage. 

The  computer  may  be  guided  by  the  following  suggestions 
and  conditions  based  on  experience.  These  suggestions  will 
tend  to  make  the  solutions  of  the  equations  less  laborious  as 
well  as  more  stable. 

1.  In  selecting  angle  equations,  preference  should  be  given  to 
the  triangles  which  have  one  or  two  sides  on  the  exterior  of  the 
figure.     This  tends  to  avoid  entanglements  with  other  conditions. 

It  is  expedient  to  exclude  triangles  with  small  angles  in  order 
to  avoid  entanglement  with  side  equations  having  large  coeffi- 
cients. It  is  often  desirable,  though  of  less  importance,  to 
exclude  triangles  that  adjoin  small  angles,  and  so  have  a  side  in 
common  with  them. 

2.  In  selecting  the  side  equations,  it  is  desirable,  as  already 
indicated  in  Art.  i  50,  to  secure  large  coef- 
ficients, and  therefore  small  angles  should 
be  used.  If,  however,  such  side  equations 
are  selected  that  the  small  angles  of  the 
figure  are  used  more  than  once,  it  may  be 
found  that  the  solution  is  unstable  be- 
cause in  some  of  the  normal  equations 
there  are  side  coefficients  of  about  the 
same  magnitude  as  the  diagonal  coefficient.  The  rule  in  select- 
ing side  equations  should  therefore  be  to  use  the  small  angles 
of  the  figure  once  and  only  once. 

3.  While  it  is  true  in  general  that  time  will  be  saved  by 
using  side  equations  having  a  small  number  of  terms,  there  are 
exceptions  to  the  rule.  For  example,  in  dealing  with  the  fol- 
lowing figure,  experience  shows  that  it  is  advisable  to  use  a  side 
equation  having  its  p(jle  at  P  and  involving  all  four  of  the  lines 
which  radiate  from  it,  and  containing  eight  terms,  although  a 
sufficient  number  of  side  equations  could  be  written,  each  of 
which  would  contain  but  six  terms.  Experience  shows  that  in 
such  a  case  as  this  the  solution   may  be  somewhat  unstable, 


Fig.  ig. 


2o6  THE    ADJUSTMENT    OF    OBSERVATIONS 

unless  this  side  equation  of  large  scope  is  used,  apparently 
because  the  points  B  and  D,  which  are  not  connected  by  a  line 
of  sight,  are  not  otherwise  sufficiently  bound  together. 

4.  In  the  process  of  eUmination,  it  is  advisable  to  avoid  the 
introduction  into  an  explicit  function  of  an  unknown  quantity 
from  which  the  corresponding  original  equation  is  free. 

In  line  4,  Table  A,  the  value  of  x  contains  the  unknown 
quantity/  which  was  absent  from  the  second  equation.  Had 
the  third  equation  changed  places  with  the  first,  the  terms  of  the 
original  second  equation  might  have  been  introduced  directly 
into  Table  A,  and  the  number  of  lines  in  Table  B  would  have 
been  two  less,  and  the  number  of  columns  one  less. 

The  order  of  solution  in  a  figure  adjustment  can  be  best 
decided  by  inspection  of  the  figure.  The  work  should  com- 
mence with  an  angle  equation  from  a  triangle  having  a  side  (or 
better,  two  sides)  on  the  exterior  of  the  figure  ;  and  no  angle 
equation  from  a  triangle  with  a  new  interior  side  should  ever  be 
introduced  till  after  the  entrance  of  every  angle  equation  not 
thus  exposed  to  entanglement  with  conditions  yet  untouched.  A 
side  equation  should  usually  be  postponed  till  after  the  introduc- 
tion of  all  the  angle  equations  that  relate  to  the  same  points  and 
no  others,  but  should  immediately  follow  them,  so  as  to  precede 
all  equations  that  extend  beyond  its  domain  into  new  territory. 

The  suggestions  in  the  preceding  paragraphs  will  be  illustrated 
in  Art,  180,  in  connection  with  the  method  of  directions. 

155.   Adjustment  of  the  Quadrilateral  NSOL  (Fig.  10). 

The  method  of  forming  the  condition  equations  having  now  been  ex- 
plained, we  are  ready  to  adjust  the  quadrilateral  NSOL,  as  promised  in 
Art.  137. 

The  condition  equations  have  all  been  formed  in  the  preceding  sections. 
Collecting  them,  we  have  : 

Local  equations  (Ex.  i,  2,  Art.  139), 

7',   +  %>.  +  7'3  =  -   1.37, 

Angle  equations  (Ex.  Art.  143), 

t'z  ^-  iu  +  7^7        =  -  048, 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       207 


Side  equation,  the  unit  being  the  sixth  place  of  decimals  (Ex.  Art.  148), 

1-43  <'i  -  0-92  ^'2  -  1-93  ^'5  +  0-74  ^6  -  0.43  "7  +  1-77  <'s  =  -  2.00. 
The   methods  of  solution  have   been  explained  in  Chapter  V,  and  we 
shall  proceed  in  the  order  there  given  for  the  three  forms. 

156.   First  Solution  —  MctJiod  of  Independent  Unknowns 

There  being  9  unknowns  and  5  condition  equations  connecting  them, 
there  must  be  4  independent  unknowns.  We  shall  choose  t/,,  v^,  7'^,  Vr,. 
Expressing  all  of  the  unknowns  in  terms  of  these  four,  we  write  the  equa- 
tions in  the  form  of  observation  equations,  as  follows  (see  Art.  118) : 


7'.=  + 
--'■1  = 

■2^6  = 


weight  2 


+ 
V,  - 


Vi  + 


+ 


+ 

7/,  — 


"^4 

+ 


-  ^-37 


7/5  +  1.07 
+  0.89 


T/g  =  —  0.565  Vi  +  0.763  v.,  —  0.661  7/4  +  0.672  7/5  —   I.361 
7/9  =  —  0.435  ^1  "~   ^-J^^}  ^2  +  0.661  7/4  —   1.672  7/5  —   1.699 

Hence  the  normal  equations 


14 

23 
6 

7 

31 
I 

8 


(I) 


"^1 

"'2 

'4 

^'5 

Const. 

+  4S.83 

+  50.70 

+  72.45 

-  32-93 

-  40.83 

+  (H-93 

+    5-44 
+  24.09 
—    2.29 
+  35-82 

-  53-45 

-  69.70 
+  28.18 

-  29.30 
+  83.79 

■=[///]■ 

Solving  these  equations  (Art.  157),  we  have  the  values  of  the  corrections, 

7/,  ==  —  0.82",  ^4  =  —  0.22", 

"^2  =  -  0.36",  '^5  =  -  0.47". 
and  thence  from  the  condition  equations, 

7/3  =  -  0.19",  v^  =-  1.33", 

^8  =  +  0.38",  Vy  =  —  0.08". 
7/7  =  —  0.07", 

These  corrections  applied  to  the  measured  values  of   the  angles  give  the 
most  probable  values  as  follows : 

M^=i24°    09'    39.87",  Afo  =  70°    39'     24.9S", 

04.71",  Mj  =  34°    40'    39-59". 

15.42",  M^  =  43°    46'    25.07", 

05.04",  M^  =  zo°    53'    30-73". 

19.94", 


M^=  113° 

39' 

^3=  122° 

11' 

M,=    23° 

08' 

M,=    47° 

3J' 

2o8  THE    ADJUSTMENT    OF    OBSERVATIONS 

157.   The  Precision  of  the  Adjusted  Values. 

(a)  To  find  the  m.  s.  e.  of  an  observation  of  the  unit  of  weight  (Arts.  105, 

108). 

From  the  above  values  of  the  residuals  77, 

[p-y-]  =  7.53- 

Check  of  [pv'^l     Carrying  through  the  solution  of  the  normal  equations  the 
extra  column  required  by  the  sum  [///],  we  find  (p.  209), 

[M']  =  7-54- 

Hence,  /^  =  i  /  '^'^'^ 

V  9-  4 

=  i  1.23". 

(l>)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  an  angle. 

Take  the  angle  A^LS.     Proceeding  as  in  Art.  108,  we  have, 

F  =  NLS 

=  180  +  e  -  {M.  +  v^  +  M^  +  7/5). 
•'.  dF  =  —  7'2  —  Vy 
Hence,  from  the  extra  column,  the  sixth,  carried  through  the  solution  of  the 
normal  equations  (p.  209), 

Up  =  0.053,  

and  therefore,  H-jr  =  i.23\/o.o53 

=  0.28". 
{c)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  a  side,  the 
base,  NS,  being  supposed  to  be  free  from  error. 
Let  us  take  the  side  OL.     We  have, 

F^  OL 

.sin  OA^S   ?,\nLSO 


=  NS 

=  NS 


sin  SON  sin  OLS 

sin  {M^  +  V3)    sin  (Mg  +  v^) 


sin  {Mj  +  Vt)    sin  (J/g  +  v^ ' 
For  check  we  shall  proceed  in  two  ways, 
(i)  Expand /^directly  ;  then, 

I^F  dF  8F  SF       \    .      „ 

=  -  0.0505  ■^3  +  0.0282  i/g  —  0.1160    7  —  0.1342  T^a 
=  —  0.007    Vx  +  O.171     V2  +  0.056       4  +  0.253    Z^5, 

by  substituting  for  v^,  v,„  Vy,  Vg,  their  values  from  equations  (i). 

Carry  through  the  solution  of  the  normal  equations  the  extra  column  re- 
quired by  these  coefficients,  and 

Up  =  0.0019. 
Hence,  fip  =  1.23  \/o.ooi9 

=  0.05  m. 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       2og 


(2)  Take  logs  of  both  members  of  the  equation  ;  then, 

log  F  =  log  .VS  +  log  sin  {M,  +  v^)  +  log  sin  {M^  +  v^) 
—  log  sin  {Afj  +  Vr)  —  log  sin  (J/g  +  Vg). 
But  since  jVS  is  constant,  we  have,  in  units  of  the  sixth  place  of  decimals, 

^/log7^  =  -  1.33  1^3 +  0.74^6  -3-04^^7  -  352  t'9 

=  —  0.18  7;,  +  4.50  Vn  +  1-45  Vi  +  6.63  z;^,  from  equations  (i). 
Hence,  from  the  last  column  added  to  the  solution  of  the  normal  equations, 

"loeF  ^  '-5°  '"  units  of  the  si.xth  place  of  decimals. 
Also,  f^iogF^  1.23  VTso 

=  1.5  in  units  of  the  sixth  place  of  decimals. 

Now,  since    </log/^  = -^=r  mod. 

and  F  =  16556  w, 

.'.  /J.F  =  0.06  m. 
The  solution  of  the  normal  equations,  with  the  extra  columns  required 
by  the  weight  determinations,  is  as  follows  : 


^1 

t'2 

''j 

7/5 

/ 

/(angle). 

/(side). 

/(side). 

+  48.83 

+  5°-7o 
+  72-45 

—  32-93 

—  40.83 
+  64.93 

+    5-44 
+  24.09 
—    2.29 

+  35-82 

—  53-45 

—  69.70 
+  28.18 

—  29.30 
+  83-79 

—  I 

—  . 

—  0.007 
+  0.171 
+  0.056 
+  0.253 

—  0.18 
+  4-50 

+  1.45 
+  6.63 

+  • 

+    1-038 
+  19.808 

—  0.674 

-  6.643 
+  42-725 

H 

H 
- 

-  0.1114 

-  18.4420 

-  1.3784 
1-35.2140 

—    1.0946 

+  14-2038 
+    7.8652 
+  23-3454 
+  25.2830 

—  I 

—  I 
0 

—  0.000 

+  0..76 

+  0.053 

+  0.253 

0 

—  0.004 

+  4.687 

+  i-328 
+  6.650 
-j-  O.OOI 

+  • 

—    0.335 
+  40.497 

+  0.931 

+  7-563 
+  .8.044 

+    0.7.76 

-j-  12.6322 
--  10.1114 
+  15.0910 

+  0.050 

—  0.335 

—  0.069 
+  0.050 

+  0.009 

+  0.112 
+  0.080 
+  0.002 

+  0.237 

+  2.900 
--2.287 

"T  >.'09 

+    • 

+  0.187 
+  16.63J 

+    0.31.2 

+    7-753 
+  "-'5' 

—  0.008 

—  0.006 
+  0.003 

+  0.003 

+  0.069 
-j-  0.000 

+  0.072 

+  ■•745 
+  0.208 

+  ■ 

[prvv.]  = 

+   0.466 
+    7.538 

—  0.000 
0 

0.000 

0.050 

0.003 

.000 

+  0.000 
0 

0.000 
0.002 
0.000 
0.000 

0.002 

=  »'/,• 

+  0.105 
+  0..83 

0.000 
1.109 
0.208 
0.183 

1.500 

=  «// 

O.OS33 

210 


THE    ADJUSTMENT    OF    OBSERVATIONS 


The  solution  has  been  carried  to  four  places  of  decimals  in  certain  parts,  on 
account  of  loss  of  accuracy  arising  from  dropping  figures  in  multiplications. 
The  resulting  values  of  the  corrections  have  been  cut  down  to  two  places  of 
decimals.  The  work  was  done  with  a  machine,  as  explained  on  p.  io6,  the 
reciprocals  of  the  diagonal  terms  being  used  so  as  to  avoid  divisions.  Thus 
the  first  reciprocal  is  0.02048. 


158.    Second  Solution  —  JMctJiod  of  Correlates. 
Arranging  the  condition  equations  in  tabular  form,  we  have 


^'1 

'■'2 

J'3 

'■'4 

^'5 

^'0 

■"1 

^'8 

Z'9 

weights  2 

2 

'4 

23 

6 

7 

31 

I 

8 

+  >-43 

+  1 

—  o.g2 
+  • 

+  .' 

+  i 

+  1 

—  i-gs 

+  ■  ' 

+  0.74 

+  1 

—  0.43 

+  1   ' 

—  I 

+  1-77 

+  1   ' 

Vi' 

—  2.00 

—  1-37 

—  0.48 

—  1.07 

—  1. 10 

TJie  Correlate  Equations. 


2  Vi  = 

2^/2  = 
14  7/3  = 
237/4  = 

67/5  = 

7  7'g  = 
31  "^'7  = 

8vl  = 


I. 

II. 

III. 

IV. 

V. 

+  1-43 

+  I 

-0.92 

+  I 
+  I 

+  I 
+  I 

+  I 

-  1-93 

+  I 

+  0.74 

+  I 

-  0.43 

+  I 

+  I 

+  1-77 

+  I 
+  I 

T/ie  Normal  Equations. 


I. 

II. 

III. 

IV. 

V. 

/. 

+  5- 

284 

+  0-255 
+  1. 07 1 

—  0.014 

+  0.071 
+  0.147 

-  0.427 

+  0.043 
+  0.353 

+  1.862 

+  0.032 
-  0.143 
+  1-300 

—  2.00 

—  1-37 

—  0.48 

—  1.07 

—  1. 10 

The  solution  of  these  equations  gives  (see  page  212) 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       211 

I.  =  -  0.3973, 
II.  =-  1.0749, 

III.  =  —  1.6006, 

IV.  =-3.5721, 
V.  =  —  0.6301. 

Substituting  these  values  in  the  correlate  equations,  the  same  values  of  the 
corrections  result  as  before.     Also, 

[pv']  =  7-53. 

159.   The  Precision. 

(a)  To  find  the  m.  s.  e.  fj.  of  an  observation  of  weight  unity. 

From  the  values  of  7'  we  find  directly, 

[^7'-]  =  7-53- 

Checks  of  [pv-].  These  are  worked  out  in  the  solution  of  the  normal 
equations  on  p.  178,  according  to  the  formulas  of  Art.  121,  and  give  7.54  and 
7.55  respectively. 

Hence,  taking  the  mean,  [pv-]  =  7.54,  and  the  number  of  conditions 
being  5, 

-\/¥ 

=  1.23",  as  before. 
Compare  Ex.  2,  Art.  122. 

(d)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  an  angle. 
Take  the  angle  NLS. 

.  ■ .  dF  =  —  V2  —  v^. 

From  the  values  of  u,  a,  b,  .  .  .  \n  the  condition  equations  in  connection 
with  the  values  of  /"given  by  this  function,  we  have 

[««/]  =+0.782,  [^4f]  =  —  0.167, 

[ubf]  = —o.soo,  ["^/]  =       o. 

[ucf]  =     o,  [»//]  =  +  0.667. 

Hence,  from  the  seventh  column  in  the  solution  of  the  normal  equations 
below, 

Uji,  =  0.053 

and  1^  p=  1.23  Vo. 053 

=  0.28". 
Compare  Ex.  4,  Art.  125. 

{c)  To  find  the  weight  and  mean-square  error  of  the  adjusted  value  of  a 
side,  the  base  being  free  from  error. 

Take  the  side  Oneota-Lester. 

As  in  {c),  Art.  157,  we  have, 

dF  =  —  0.0505  7;.,  +  0.0282  ?;„  —  0.1160^7  —  0.13425  x'u- 
Also  from  the  condition  eciualions, 


212 


THE  ADJUSTMENT  OP  OBSERVATIONS 


[  uaf^  =  +  0.0046,  [  udf^  =  -  0.0040, 

[  "i-'f\  =  -  0.0036,  [  uef'\  =  -  0.0165, 

[  1":/  J  =  -  0.0073,  [  ?(^]  =  +  0.0030. 

Hence,  from  the  eighth  column  in  the  solution  of  the  normal  equations, 

My  =  0.0023, 

and  finally,  I>-f  =  ^-^j  V.0023 

=  0.06  771. 

Solution  of  the  u\or»ial  Equations. 


+  5.284 


+  1 


II. 


+  0.2S5 
+  1. 071 


+  0.0483 
+1.0587 


+  1 


—  0.014 
+  0.071 
+  0.147 


IV. 


+  0.043 
+  0.353 


.0026       — o.oS 


+  0.0717 
+  0.T470 


+  0.0677 
+  o. 1421 


+  1 


+  0.0206 
-f  0.0419 

+0.3185 


+  0.0204 
+  0.0404 

+  0.3I8I 


+  0.2843 
+  0.3066 


J''alues  of  the  Unknmims  : 

1.  =  — 0.3973- 

II.  ;=  —  T  .0749, 

Ill.rr—  1.6006, 

IV.  =  -3. 5721, 

V.  rr  — o.  6301. 


+  1 


V. 


+  1.862 

+  0.032 
—  0.143 
+ I . 300 


+  0.3524 

—  0.0902 
+  0.0369 
+  0.007s 

+  0.6438 


—  0.0851 

+  0.0434 
+  0.0093 
+  0.6361 


+  0.3054 

—  0.0030 

+  0.6228 


—  0.009a 

+  0.6228 


+  1 


y(ANGLE)  _/(SiDe) 


1.07 
I.  10 


+0.782 
0.500 


—  o. 167 
+  0.667 


0.3785 

I. 2731 

0.4853 

—  1. 2316 

'■3952 

+  0.7570 


+  0.1 480 
—  0.5377 

+  O.O02I 
>.  1035 
5.2756 

+  0.5510 


—  0.5037 
+  1.5309 


—1.0933 

—  0.3818 
+  I  . 1209 


—  3.5650 


—  0.3924 
+3.8986 


—  0.6301 

=  V 

+  0.  2472 


I.  X  /'    =  —  0.3973  X  —  2.00  =  0.7Q 
II.  X  /"   =  —  1.0749  X  —  1.37  =  1-47 

III.  X  /"'  =—  1.6006  X  —0.48  =  0.77 

IV.  X  /""  =  —  3-5721  X  —  1.07  =  3  82 
V.  X  /'""  r=  —  0.6301  X  —  i.io  =  o.  69 

7.54 


0.7570 
I . 5309 

1. 1 209 

3.8986 

0.2472 


7  5546 


+  0.0046 

—  0.0036 
0.0073 

—  0.0040 
0.0165 

+  0.0030 


+  o . 0009 

0.0038 
0.0073 

—  0.0036 

—  0.0182 


-0.5078 

+  0.0385 
—  0.0930 
3.3214 
+  0.2780 


—  0.0036 

—  0.0070 

—  0.003s 

—  0.0185 
+  0.0630 


+  0.  2709 

—  o. 1030 

—  0.3332 

+  0.2676 


—  0.0015 

—  0.0164 

+  0;0027 


-0.3389 

—  0.3342 
+  0.2324 


—  0.0164 
+  0.0027 


—  0.5366 
+  0.0531 


—  0.0264 
+  0.0023 


u, 


The  values  of  Ipv'']  are  found  from  Equations  2,  3,  Art.  121. 


ADJUSTMENT   OF   A    TRIANGULATION  —  ANGLES       213 

160.    Third  Solution  —  Solution  in  Tzvo  Groups. 
The  form  given  in  Art.  128  is  followed. 

The  Local  Adjustment. 
{a)  At  North  Base. 

The  Observation   Equations. 


p 

(-Tl) 

(A^ 

/ 

2 

+   I 

0. 

2 

.    .    . 

+   I 

0. 

14 

—   I 

—   I 

-  1-37 

The  Normal  Equations. 

16  +  14  =  —  19.18  =  \,par\  suppose, 
14  +  16  =  —  19.18  =  \_pbl\  suppose. 
Solving  in  general  terms, 

{X,)  =+0.267  \pal^  -0.233  {pbl\ 
(jr,)  =  -  0.233  lPal\  +0-267  {pbl\ 


Hence, 


and 


(j-,)  =  -  0.64", 
{x^)  =  -  0.64", 

(^3)  =+ 0.64"  +  0.64"  =  1.37", 
=  -  0.09", 


Local  Angles. 
124°   09'   40-05", 
113°   39'   04.43", 
122°   11'   15.52". 

To  find  the  m.  s.  e.  of  a  single  observation. 
The  value  of  \pv'^'\  =  [pxx]=  1.75. 
Hence,  for  this  station,  the  number  of  conditions  being  3 

-75 


-^J^ 


=  13" 

(d)   At  South  Base. 

The   Observation    Equations. 


/ 

(X,) 

(^J 

/ 

23 

+  I 

0. 

6 

+  I 

0. 

7 

+  I 

+  I 

-'■37 

214  THE    ADJUSTMENT    OP    OBSERVATIONS 

The    Normal   Equations. 

30  +    7  =  -  7-49. 
7  +  13  =  -  7.49- 
Hence 


Also, 


(x,)  =  -  0.13", 

(Xg)  =  -  0.50", 

(;t-J   =    _  0.13"  -0.50"  + 

1.07     , 

=    +  0.44". 

Local  Angles. 

23^     08'      05.13", 

47°    31'     19-91". 

70°    39'     25.04". 

[^7/^]  =  [pxx]  =  3.24. 

•••-v^^-^^ 

=  1.8". 

The  General  Adjustment. 
Mosf  Probable  Angles. 


At  N.  Base, 

124° 

113° 

122° 

09' 

39' 
11' 

40.05" 

04-43" 
15.52' 

'+(1), 
■  +  (2), 
'  -  (I)  -  (2). 

At  S.  Base, 

23° 
47° 

70° 

08' 
31' 
39' 

05-13" 
19.91" 
25.04" 

■  +  (4), 
'  +  (5), 

■  +  (4)  +  (5)- 

At  Oneota, 

34° 
43° 

40' 
46' 

39-66" 
26.40" 

'  +  (7), 
'  +  (8). 

At  Lester, 

30° 

53' 

30.81" 

'  +  (9). 

The  Angle  and  Side  Equations, 
(a)  Triangle,  N.  Base,  S.  Base,  Oneota. 
Angle  SNO  122°  11'  15.52"  -  (i)  -  (2) 
"      JVSO    23°  08'  05.13"  +  (4) 
"      ATOS    34°  40'  39-66"  +  (7) 

Sum     =  180°  00'  00.31" 
180  +  €  =  180°  00'  00.05" 


o  =  0.26"  -  (I)  -  (2)  +  (4)  +  (7) 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       215 

{b)  Triangle  Lester,  Oneota,  S.  Base. 
Angle  NSO  70°  39'  25.04"  +  (4)  +  (5) 
SOL  78°  27'  06.06"  +  (7)  +  (8) 
"        OLS  30°  53'  30.81"  +  (9) 


iSo    00   01.91 
180°  00'  00.37' 


o  =  1.54"  +  (4)  +  (5)  +  (7)  +  (8)  +  (9) 
{c)  Quadrilateral  N.  Base,  S.  Base,  Oneota,  Lester. 
sin  LNS  sin  LSO  sin  LON 
sin  LNO  sin  NSL  sin  LOS  ~^' 


LNS  =  113°  39'  04.43"  +  (2),  LNO  =  124°  09'  40.05"  +  (i), 

LSO  =    70^  39'  25.04"  +  (4)  +  (5),  NSL  =    47°  31'  19-91"  +  (5), 

LON=    43°  46'  26.40"  +  (8),  LOS  =    78°  27'  06.06"  +  (7J  +  (8). 

9.9618975,6  -    9,22  (2)  9.9177479,3  -  14,29  (i) 

9.9747660,1  +    7,39  K4)  +  (5)  I  9.8677849,8  +  19,28  (5) 

9-8399903>4  +  21,98  (8)  9.9911180,3  4-    4,3©  {(7)  +  (8)  \ 


539'!  509,4 

509,4 


29,7 


Check  by  deducting  \  of  the  spherical  excesses  of  the  triangles  from  the 
angles. 

"3°  39'  04.36",  124°  09'  40.01", 

70°  39'  24.92",  47"  31'  19-84", 

43^  46'  26.36",  78°  27'  05.93". 

9.9618976.2  9.9177479,9 

9.9747659.3  9.8677848,6 
9.8399902,5  9.9911179,8 

M 
29,7 

The  two  methods  agree  well. 

A  glance  at  the  log  differences  for  i"  shows  that  by  expressing  them  in 
units  of  the  sixth  place  of  decimals  their  average  value  is  unity  nearly.  We 
have,  then,  for  the  side  equation, 

1.43  (ij  -  0.92  (2)  +  0.74  (4)  -  1. 19  (5)  -  0.43  (7)  -I-  1.77  (8)  +  2.97  =  o. 


2l6  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  Weight  Equations. 
(i)  =- 0.233  |T|  +0.267(7] 
(2)  =+  0.267  [Tj  -0.233  0 
(4)  =  +  0.038  (4]  -  0.021  ui 

(5)=  -0.021  (2  +0.088  0 

(7)  =  +  0.032  0 

(8)  =  +    1. 000  [8] 

(9)=  +0.125  0 

The  Correlate  Equations. 

I.                II.                     III.  Check. 

B  =-  I                                 +  1-43  =  0.43 

[2]  =  —  I                                  —  0.92  +  1.92 

ITI  =  +  I         +1         +  0.74  -  2.74 

B  =             +1         - 1. 19  +  0.19 

[7]  =+  I         +1         -  0.43  -  1.57 

[8;i  =           +1        + 1.77  -  2.77 

[2^1  =           +1  —  1. 00 

The  check  is  formed  by  adding  each  horizontal  row  (Art.  78). 

Expression  of  the  Corrections  in   Terms  of  the  Correlates. 

I.                              II.                          III.  Check. 

+  0.233                                               -  0.333  +0.100 

—  0.267                                               —  0.246  +  0.513 

(1)  =     -  0.034                                               -  0.579  +  0.613 

—  0.267                                       +0.382  —O.I  15 

+  0.233                            +  0-214  -  0.447 

(2)  =    —  0.034                                     +  0.596  —  0.562 

+  0.038             +  0.038             +  0.028  —  0.104 

—  0.021              +  0.024  —  0.004 


(4)  =    +  0.038  +  0.017  +  0.052  —  0.108 


0.021  —0.021  —0.016  +0.058 

+  0.088  —0.105  +0.017 


- 

0.579 

+ 
+ 

0.382 

0.214 

+ 

0.596 

+ 

0.028 

+ 

0.024 

+ 

0.052 

— 

0.016 

- 

0.105 

(5)    =  —  0.021                   +  0.067  —  O.I2I  +  0.075 

(7)  =  +  0.032                   +  0.032  —  0.014  —  0.050 

(8)  =                            +1.  +  1.770  —  2.770 

(9)  =                        +  0.125  —  0-125 


ADJUSTMENT   OF    A    TRIANGULATION  —  ANGLES       217 
The  Corrections  in  Ter>ns  of  the  Correlates  {Collected). 


I. 

II. 

III. 

(I)  = 

-  0.034 

-  0.579 

(2)   = 

-  0.034 

+  0.596 

(4)   = 

+  0.038         + 

0.017 

+  0.052 

(5)  = 

—  0.021         + 

0.067 

—  0.121 

(7)  = 

+  0.033         + 

0.032 

—  0.014 

(8)  = 

+ 

1. 000 

+  1.770 

(9)  = 

+ 

0.1 

-5 

JFor/nation  of  the  A'or/zial 

Equations. 

I. 

II. 

in. 

Check. 

-  (i)  =    +  0.034 

+ 

0.579 

—  0.613 

-  (2)  =    +  0.034 

- 

0.596 

+  0.562 

+  (4)  =    +  0.03S 

+  0.017 

+ 

0.052 

—  0.108 

+  (7)  =    +  0-032 

+  0.032 

- 

0.014 

—  0.050 

+  0.138 

+  0.049 

+ 

0.021 

+  0.209 

(4)  = 

+  0.017 

+ 

0.052 

—  0.  loS 

(5)  = 

+  0.067 

- 

O.I2I 

+  0.075 

(7)  = 

+  0.032 

- 

0.014 

—  0.050 

(8)  = 

+  I. 

+ 

1.770 

-  2.770 

(9)  = 

-f  0.125 
+  1.241 

—  0.125 

+  0.049 

T 

1.687 

-  2.978 

I.                  II. 

III. 

Check. 

+  1-43(0 

+  0.852 

—  0.804 

-  0.92(2) 

+  0.533 

-  0.564 

+  0.74(4) 

+  0.038 

—  0.080 

-  1-19(5) 

+  0.144 

—  0.089 

-  0.43(7) 

+  0.006 

+  0.022 

+  1-77(8) 

■^  3-133 
+  .1.706 

-  4903 

+ 

0.021         +  1.687 

-  6.418 

The  A'cr/nal  Equations  {Collected). 

I.  II.  III. 

+  0.138         +  0.049         +  0021  =—  0.260 
+  1. 241  +  1.687  =—  1-540 

+  4-706  =  -  2.970 
The  .solution  of  these  equations  gives  (page  220) 

I.  =-1.597 

II.  =  —  0.642 

III.  =-  0.394 

Substitute  for  I.,  II.,  III.,  their  values  in  (4),  and  we  have  the  general 
corrections. 


2l8 


THE    ADJUSTMENT    OF    OBSERVATIONS 


Adding  the  local  corrections  and  general  corrections  together,  the  total 
corrections  to  the  measured  angles  result  and  are  as  follows : 


Local. 

General. 

Total. 

/ 

Pia 

Final  A 

NGLES. 

x\  = 

—  0.64 

-  0.18 

=  -0.82 

2 

1-34 

0 
124 

09 

3987 

^2  = 

—  0.64 

+  0.28 

=  —  0.36 

2 

.26 

"3 

39 

4.71 

^3  = 

—  0.09 

—  O.IO 

=  —  0.19 

14 

.50 

122 

II 

15.42 

^4  = 

-0.13 

—  0.09 

=  —  0.22 

23 

1. 10 

23 

08 

5.04 

^5  = 

—  0.50 

+  0.04 

=  —  0.46 

6 

1.27 

47 

31 

19.95 

^6  = 

+  0.44 

—  0.05 

=  +  0.39 

7 

1.06 

70 

39 

24.99 

^7  = 

.... 

—  0.07 

=  —  0.07 

31 

•15 

34 

40 

39-59 

•^8  = 

-  1-33 

=  -  1-33 

I 

1.77 

43 

46 

25.07 

^9  = 

—  0.08 

=  -  0.08 

8 

.05 

30 

53 

3073 

[pv^ 

=  7-5° 

Number  of  local  conditions       =  2 

Number  of  general  conditions  =  3 

Total     =  5 

The  method  of  solution  just  given  is  substantially  the  same  as  that  em- 
ployed on  the  survey  of  the  Great  Lakes  between  Canada  and  the  United 
States  by  the  U.  S.  Engineers. 

161.   The  Precision  of  the  Adjusted  Values. 

{a)  To  find  the  m.  s.  e.  of  an  observation  of  weight  unity. 

Computation  of  [^i'-]. 

(i)  From  the  preceding  table  \pv^'\  has  been  found  directly;  thus, 

[^7;2]  =  7.50. 

(2)  Check  (Art.  129).     From  the  station  adjustments  find  [7'°?/°]. 

N.  Base  gives  (p.  213)     1.75 
S.  Base  gives  (p.  214)     3.24 

4.99  =  b°'z^°]- 

From  the  general  adjustment  find  \t.uiv\ 


(«)  i: 
I" 

L" 


(i3)   /o' 


X         I. 

X     II. 
X  III. 


X   p^ 


X 


[fe. 


=  —  0.26  X  —  1.597  =  +  0.42 
=  —  1.54  X  —  0.642  =  +  0.99 
=  —  2.97  X  —  C.394  =  +  1. 16 

=  —  0.26  X  —  1.885  =  +  °-49 

r=  -     1.45     X    -     1.183    =  +     1.72 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        219 


and 


C.2    X  [^?=z^=  -  0.94  X  -  0.394  =  +  0.37 

UC.2J  ^ 

+   2.58 

.'.  [ww]      =  2.58 
Ipv']  =  4-99  +  2.58 
=  7-57. 


Hence,  taking  the  mean  of  the  values  of  [pv'], 


there  being  2  local  conditions  and  3  net  conditions. 

{d)  To  find  the  m.  s.  e.  of  an  angle  in  the  adjusted  figure. 

Angle  =  JVLS. 

.-.    ./i^  =  -(2)-(5) 

= +0.055 1-  ~  0.067  n.  +  0.700  III. 

from  the  weight  equations. 

From  equations  (25),  Art.  130, 

?i  =  [aa],?'i  +  [«^]i"2  +  •  •  • 

?2  =  [«i3]^.  +  im^r^  +  •  •  • 


The  values  of  [aa],  [ajS]  .  .  .  are  given  in  the  vi^eight  equations.     Hence, 

gi=+  0.267  X  o  -  0.233  X  -  I  =  +  0.233, 

^2  =  —  0.233  X  o  +  0.267  X  —  I  =  —  0.267, 

^3  =  +  0.038  X  o  —  0.021  X  —  I  =  +  0.021, 

q^=  —  0.021  X  0-+  0.088  X  —  I  =  —  0.088. 


g 

(I 

g<l 

0 

+  0.233 

0 

—  I 

—  0.267 

0.267 

0 

+  0.021 

0 

—  I 

-  0.088 

0.088 

Ua] 

[^B.lP 


(See  the  solution     [ffc]]  =      0.355 


of  the  normal 

equations.) 


[C  C.2] 


0.006 


=  0.274 
0.302 


~  °:3°t 
7//,.  =+  0.053 

IX  p=  i.23"\/o.o53 
=  0.28". 


220 


THE    ADJUSTMENT    OF    OBSERVATIONS 


{c)  To  find  the  m.  s.  e.   of  a  side  in  the  adjusted  figure. 
Side  =  Oneota-Lester. 


F=  0L  =  NS 


sin  ONS   sin  OSL 
sin  SON  sin  OLS' 


Therefore, 

dF=  1.33  (I)  +  1.33  (2)  +  0.74  (4)  +  0.74(5)  -  3-04  (7)  -  3-52  (9) 

in  units  of  the  sixth  place  of  decimals, 

=  —0.174  I-  —  0.475  ^  I-  +  0.015  I^I- 

from  the  weight  equations. 

The  solution  is  carried  through  exactly  as  in  the  preceding  case.     We 
find, 

{gq\  =  2.01 1    and    n^  =  1.49. 


Hence, 

Now, 


fiVu^^  i.23\/i.49 

=  1.5  in  units  of  the  sixth  place  of  decimals, 
log  OL  =  4.2189699         OL  =  16556  m. 

r       -J  16556  ■ 

m.  s.  e.  of  side  =  — ^^-^^  x  0.0000015 
0.434 

=  0.06  m. 


Solution  of  the  N^on/ial  Equations. 


I. 

11. 

III. 

/ 

/"(Angle). 

+  0.138 

+  0.049 
+  I. 241 

+  0.021 
+  1.687 
+  4706 

-  0.260 

-  1.540 

-  2.970 

+  0.055 
—  0.067 
+  0.700 

+  I 

+  0-355 
+  1.224 

+  0.152 
+  1.680 
+  4-703 

-  1.885 

-  1.448 

-  2.930 

+  0.399 
—  0.087 
+  0.692 
+  0.022 

+  I 

+  1-373 
+  2.396 

-  1-183 

-  0.945 

—  0.071 
4-  o.Sii 
+  0.006 

+  I 

-  0-394 

+  0.338 
+  0.274 

ADJUSTMENT    OF    A    TRIAXGULATION  —  ANGLES        221 


162.     Ex.  I.  —  Adju.st  the  observed  differences  of  longitude  *  given  in 
the  following  table : 

HEARTS  CONTENT  TOILHOMMERUM 


Fig.  20. 


'WASHINGTON 


Dates. 

Observed  Differences. 

Correc- 
tions. 

1851 

1857 
1866 
1866 
1866 
1872 
1872 
1872 
1872 
1872 
I 869-1 870 
1870 
1867  ( 
1872) 
1872 

Cambridge-Bangor 

Bangor-Calais 

Calais-Heart's  Content     .     .     . 
Heart's  Content-Foilhommerum 
Foilhommerum-Greenwich   .     . 

Brest-Greenwich 

Brest-Paris 

Greenwich-Paris 

St.  Pierre-Brest 

Cambridge-St.  Pierre  .... 
Cambridge-Duxbury    .... 
Duxbury-Brest 

Washington-Cambridge   .     .     . 

Washington-St.  Pierre     .     .     . 

h.       tn.          s.                 s. 
0        9      23.080  ^  0.043 

6    00.316^:30.015 
55    37-973  it  0.066 

2  51      56.356  rt  0.029 
41      33-336^0.049 
17      57.598  zt  0.022 
27      18.512  ^  0.027 

9      21.000  J- 0.038 

3  26     44.S10 -[- 0.027 
59     48.608^^0.021 

I      50.191  ^  0.022 

4  24     43.276^0.047 

23     41.041  J;  0.018 
23      29.553  ±0.027 

v.. 

[Number  of  conditions  =  ;;  —  j  +  i,  where  n  is  number  of  observed  dif- 
ferences of  longitude,  and  s  is  number  of  longitude  stations. 
The  condition  equations  are  s. 

—  T/g   +  v^   —  7/g    =  +  0.086 
-  Vi-  v^-  V3-  v^-  v^  +  Va  +7/9   +  ^'lo  =  +  0.045 

-  Z/g  -  T/jo  +  -Z^ii  +  -2^12  =    -  0.049 
+   ^10  +    ^13  -   ■Z^M   =    -   0096 


The  weights  are  taken 
inversely  as  the 
squares  of  the  p   e. 

Solution  by  method 
of  correlates,  as  in 
Art.  119.] 

Ex.  2.  —  The  system 
of  triangulation  shown 
in  the  figure  was  exe 
cuted  by  Koppe  in  the 
determination  of  the 
axis  (Airolo-Gos- 
chenen)    of     the     St. 


vm 


xn 


Fig.  21. 


*  Coast  and  Geodetic  Survey  Report,  1880,  app.  No.  6. 


222 


THE  ADJUSTMENT  OF  OBSERVATIONS 


Gothard  tunnel*  In  the  following  table  the  adjusted  values  are  given 
side  by  side  with  the  measured  values.  It  is  proposed  as  a  problem  of 
adjustment. 


At  Goschenen. 

II. 

0 

O 

III. 

44 

IV. 

69 

V. 

124 

At  II. 

III. 

0 

IV. 

37 

V. 

60 

VI. 

77 

Goschenen 

93 

VII. 

124 

At  III: 

VIII. 

0 

IX. 

53 

VI. 

99 

IV. 

102 

Goschenen 

138 

VII. 

144 

II. 

180 

At  VIII. 

XI. 

0 

XII. 

18 

X. 

43 

IX. 

50 

VI. 

106 

V. 

112 

III. 

130 

At  IX. 

VI. 

0 

V. 

8 

III. 

18 

VIII. 

63 

X. 

76 

XI. 

79 

Airolo 

109 

XII. 

123 

Measured.    Adjusted. 
/  //  // 

00      00.00      00.00 

^T,  10.88  10.03 
30  12.51  11.62 
58  4.23  5.13 


00  00.00  00.00 

53  54-33  52-97 

29  33-'^3  33-82 
4  5-67  8.17 

II  41.69  40.57 

16  33-98  33.27 

00  00.00  00.00 

58  14-48  15-49 
47  50-21  50.86 

32  51-36  51-90 

44  28.81  29.70 
28  12.47  11.40 

59  38.94  39-11 

00  00.00  00.00 

56  17.43  17.54 

50  24.03  24.70 

18  22.52  20.27 

30  15.04  15.37 
28.72  29.24 

II  30.81  41.54 

00  00.00  00.00 

28  17.13  15.06 

33  3-27  5-00 
41  28.63  28.55 
59  50-89  51-48 
10  36-33  36-34 

45  39-23  39-33 
16  23.76  24.23 


At  IV. 


V. 
VI. 
VII. 
Goschenen 
II. 
III. 
At  V. 
IV. 
VIII. 
IX. 
VI. 
VII. 
Goschenen 
II. 
At  VII. 
II. 
III. 
IV. 
V. 
VI. 
At  XI. 
XII. 
Airolo 
IX. 
VIII. 


At  XII. 

IX. 

Airolo 

X. 

VIII. 

XI. 


At  Airolo. 
XI. 


Measured.  Adjusted. 

o  /  //  // 

o  00  00.00  00.00 

15  41  3-57  6.29 

74  12  20.55  19.86 

80  32  48.99  50.12 

135  44  49.77  50.91 

199  24  11.56  10.73 


78 
140 

215 
286 
316 
338 


00 
40 
44 
32 
19 
00 
20 


00.00 
5-91 
43-51 
45-41 
25-30 
44.92 

33-53 


00.00 
6.72 
44-45 
43-45 
27.21 
43.61 
31-74 


o  00  00.00  00.00 

19  II  58.44  59.03 

32  4  49.32  48.68 

64  II  54.08  56.05 

90  05  39.47  37.00 

o  00  00.00  00.00 

16  55  55-o6  54.38 

37  13  59-79  58-43 

152  26  30.24  30.44 


o  00  00.00  00.00 

30  31  2.30  3.39 

42  13  20.53  21.33 

90  3  2.22  1.74 

98  40  14.95  13-72 


00     00.00     00.00 


*  Zeitschr.  fiir  Veriiiess.^  vol.  iv. 


ADJUSTMENT    OF    A    TRIAXGULATION  —  ANGLES        223 


AtX. 

XII. 

94 

54 

56.06     55.26 

XII. 

0 

00 

00.00 

00.00 

IX. 

230 

53 

7.51       6.98 

Airolo 

9 

49 

30.02 

37-92 

X. 

296 

26 

49-43     51-" 

IX. 

91 

30 

5.16 

5-96 

VIII. 

252 

43 

46.75 

47-49 

XI. 

275 

12 

8.44 

9-74 

The  distance  X-XII  is  4416.S  )n. 

There  are  19  angle  equations  and  15  side  equations  in  the  adjustment. 

Solution  by  Groups. 

163.  The  rigorous  forms  of  solution  which  have  been  given 
are  suitable  for  a  primary  triangulation  where  the  greatest 
accuracy  is  required.  In  secondary  or  tertiary  work  it  is  fre- 
quently not  advisable  to  spend  so  much  labor  in  the  reduction. 
For  work  of  this  kind  the  group  method  of  solution  is  to  be 
preferred. 

The  solution  by  groups  may  be  made  by  either  of  two  general 
methods.  First,  each  condition  or  set  of  conditions  may  be 
adjusted  for  independently  in  succession,  the  values  of  the 
corrections  found  at  each  adjustment  being  closer  and  closer 
approximations  to  the  final  values.  Should  the  values  found, 
after  going  through  all  of  the  conditions,  not  satisfy  the  first 
and  second  groups  of  condition  equations  closely  enough,  the 
process  must  be  repeated  until  the  required  accuracy  is  attained. 
This  is  the  method  outlined  in  general  terms  in  Art.  131. 

Second,  each  group  may  be  adjusted  in  turn  while  the  results 
of  the  adjustment  of  the  preceding  groups  are  preserved  by 
insuring  that  the  conditions  which  have  been  satisfied  remain  so. 
This  is  an  exact  method. 

164.  To  make  the  operation  as  simple  as  possible,  let  us  take 
but  a  single  condition  at  a  time. 

(i)   Local  equation  at  N.  Base, 

7'i  +  v.^  -H  z's  +  1.3 7  =  o- 
The  soluti(jn  is  given  in  Ex.  i,  Art.  139, 

V^  =   —  0.64",    V.^  =    —  0.64",    7'3  =    —  0.09. 


224  THE    ADJUSTMENT    OF    OBSERVATIONS 

(2)  Local  equation  at  S.  Base, 

7'4  +   5"5— 5-6   +    1.07    =  O. 

The  solution  is  given  in  Ex.  2,  Art.  139, 

Z\   =    -   0.13",   7-5  =    -   0.50",   7'6  =    +   0.44". 

(3)  Angle  equation, 

7'3  +  7'4  +  7/7  +  0.48  =  o. 

Using  the  values  of  v^,  v^,  already  found  as  first  approxima- 
tions, the  equation  reduces  to 

^'3  +    7'4+   ^'7  +   0-26    =    O. 

The  method  of  solution  is  given  in  Ex.  2,  Art.  120, 

t'3  =  —  0.13",  ?'4  =  —  0.08",  z;  =  —  0.05". 
The  successive  approximations  found  so  far,  when  added,  give 

t\  =  -  0.64",  Vr,  =  -  0.55", 

V,  =  -  0.64",  z'e  =  +  0.44", 

7-3  =  —  0.22",  Z't  =  —  0.05"- 

7'4  =    —  0.21", 

Proceed  similarly  with  the  remaining  two  condition  equations. 
The  resulting  values  will  agree  closely  with  the  rigorous  values 
already  found. 

165.  In  order  to  bring  out  still  more  clearly  the  advantages 
of  solving  in  this  way,  let  us  take  a  more  extended  example. 
A  good  one  is  furnished  by  the  triangulation  (1874-1878)  of 
the  east  end  of  Lake  Ontario,  omitting  the  system  around  the 
Sandy  Creek  base. 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES        225 


Fig.  22. 


The  measured  values  of  the  angles  are  given  in  the  following 
table.  Each  angle  is  taken  to  be  of  the  same  weight.  In  the 
last  column  are  given  the  locally  corrected  angles  found  by  the 
rigorous  methods  of  solution. 


Station- 
Occupied. 

Angle  as  Measured  Between 

L(>C.\LLY 
COKK. 

Angles. 

Sir  John  .    . 

Carlton  and  Kingston    .    . 

0 
90 

17 

44.91 

" 

Wolfe  and  Kingston. 

56 

24 

09.77 

Carlton     .    . 

Wolfe  and  Sir  John    . 
Kingston  and  Sir  John 

120 
62 

48 
03 

06.54 
27.56 

Kingston.    . 

Sir  John  and  Wolfe   . 
Carlton  and  Wolfe     . 
Wolfe  and  Amherst  . 

64 

7,1 
88 

40 
02 
19 

50.91 

04  •  43 
14.70 

Wolfe  .    .    . 

Duck  and  Carlton  .    . 
Amherst  and  Carlton 
Kingston  and  Carlton 

1 88 
140 
84 

07 
I  2 

18-54 
34-44 
•4-34 

Sir  John  and  Carlton 

25 

18 

16.80 

226 


THE  ADJUSTMENT  OF  OBSERVATIONS 


■ — 

Locally 

Station 

Angle  as  Measured  Between 

CORR. 

Occupied. 

Angles. 

Amherst  .    . 

Kingston  and  Wolfe.    .    . 

o 

35 

41 

23.02 

// 

22  .69 

Kingston  and  Duck  .    .    . 

III 

45 

28.46 

28.68 

Wolfe  and  Duck    ...    . 

76 

04 

06.32 

05  ••99 

Grenadier  and  Duck.    .    . 

54 

3^ 

00.34 

Duck  and  Vanderlip.    .    . 

71 

15 

25-43 

25-32 

Vanderlip  and  Kingston  . 

176 

59 

06.11 

06.00 

Duck    .    .    . 

Oswego  and  Vanderlip.    . 

104 

08 

58.93 

59.10 

Vanderlip  and  Amherst    . 

70 

26 

31-99 

32.16 

Amherst  and  Wolfe  .    .    . 

56 

01 

12.47 

12.64 

Wolfe  and  Grenadier    .    . 

18 

45 

43-36 

43-53 

Grenadier  and  Stony  Pt.  . 

49 

53 

12.77 

12.94 

Stony  Pt.  and  Oswego  .    . 

60 

44 

19.46 

19.63 

Grenadier    . 

Stony  Pt.  and  Duck  .    .    . 

78 

13 

33-64 

33-84 

Duck  and  Amherst    .    .    . 

50 

35 

04.28 

04.19 

Duck  and  Stony  Pt.  .    .    . 

281 

46 

25-89 

26.16 

Amherst  and  Stony  Pt.     . 

231 

II 

22 .04 

21.97 

Stony  Point. 

Oswego  and  Duck .... 

88 

22 

00.86 

.    . 

Duck  and  Grenadier.    .    . 

51 

53 

12 .60 

12   70 

Grenadier  and  Duck.    .    . 

308 

06 

47.21 

47-3° 

Oswego    .    . 

Sodus  and  Vanderlip    .    . 

80 

29 

46  10 

46-59 

Sodus  and  Duck     .... 

107 

19 

03.28 

03.96 

Sodus  and  Stony  Pt. .    .    . 

138 

12 

49.44 

48.28 

Vanderlip  and  Duck.    .    . 

26 

49 

16.61 

17-37 

Vanderlip  and  Stony  Pt. . 

57 

43 

01 .96 

01 .69 

Duck  and  Stony  Pt.  .    .    . 

30 

53 

42.88 

44-32 

Vanderlip    . 

Amherst  and  Duck    .    .    . 

38 

18 

07  .12 

07.30 

Amherst  and  Oswego    .    . 

87 

19 

53-47 

53-i6 

Duck  and  Oswego.    .    .    . 

49 

01 

45-54 

45-86 

Duck  and  Sodus    .... 

87 

59 

12.55 

12.42 

Oswego  and  Sodus    .    .    . 

38 

57 

26.55 

26.56 

Sodus  and  Amherst  .    .    . 

233 

42 

40.41 

40.28 

Sodus   .    .    . 

Vanderlip  and  Oswego.    . 

60 

32 

57-55 

The  local  and  general  equations  are  formed  as  usual  (see 
Arts.  {133-155).  The  general  rule  in  the  solution  is  to  adjust 
for  one  condition  at  a  time.  Instead,  however,  of  following  out 
this  rule  strictly,  it  is  often  better  to  adjust  for  a  £-ro?ip  of  con- 
ditions simultaneously.  Often  a  group  is  almost  as  easily 
managed  as  a  single  condition.  No  rule  can  be  given  to  cover 
all  cases,  and  much  must  be  left  to  the  judgment  and  ingenuity 
of  the  computer. 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES     227 


166.   The  Local  Adjustment  at  Each  Station. 

[a)  Adjust  for  each  sum  angle  separately. 
Rule  and  example  in  Arts.  138-139. 

(b)  Adjust  for  closure  of  the  horizon. 
Rule  and  example  in  Arts.  138-139. 

At  stations  Sir  John,  Carlton,  Kingston,  Wolfe,  there  are  no 
local  conditions  ;  and  at  each  of  the  stations  Amherst,  Stony 
Point,  Sodus,  there  is  one  angle  independent  of  the  others,  and 
therefore  not  locally  adjusted. 

The  angles  at  station  Amherst  may  be  rigorously  adjusted, 
as  in  Art.  138.  The  resulting  values  are  given  in  the  table.  If 
we  break  the  adjustment  into  two  parts,  as  in  [a)  and  [b),  we 
have  : 

{a)   Sum  Angle. 


0 

Measured  Values. 

Adjusted. 

Kingston-Wolfe, 

35 

41 

23.02  —  0.29 

22.73 

Wolfe-Duck, 

76 

04 

06.32  —  0.29 

06.03 

III 

45 

29-34 

28.76 

Kingston-Duck, 

III 

45 

28.46  +  0.29 

28.75  check. 

3)0.88 

0.29 

Closure  of  Horizon. 

0 

// 

Kingston-Wolfe, 

35 

41 

22.73  ~  °-°^ 

22.65 

Wolfe-Duck, 

76 

04 

06.03  ~  °-°7 

05.96 

Duck-Vandeiiip, 

71 

15 

25.43  —  o.oS 

25-35 

Vanderlip-Kingston, 

176 

59 

06.11  —  0.07 
4 )  00.30 

06.04 

00.00  check. 

00.075 

The  adjusted  values  agree  closely  with  those  from  the  simultaneous 
solution,  as  given  in  the  table. 

At  station  Duck  the  angles  close  the  horizon.  Hence  the 
correction  to  each  angle  is  one-sixth  of  the  difference  of  their 
sum  from  360'^.      (See  Art.  139.) 

167.  The  General  Adjustment. — The  local  adjustment 
being  finished,  we  .shall  consider  the  adjusted  angles  to  be  inde- 


22S  THE    ADJUSTMENT    OF    OBSERVATIONS 

pendent  of  one  another  and  to  be  of  the  same  weight.  We  are 
therefore  at  hberty  to  break  up  the  net  into  its  simplest  parts. 
We  have  in  our  figure,  first  a  quadrilateral  SCWK,  next  two 
single  triangles  KWA,  A  WD,  next  a  central  polygon  DAGS 
and,  lastly,  a  single  triangle  VOS.  These  three  figures  include 
most  cases  that  arise  in  any  triangulation  net. 

1 68.  {a  i)  Adjustment  of  a  Quadrilateral:  Approximate 
Method.  —  In  the  quadrilateral  SCK  W  all  of  the  eight  angles 
I,  2,  .  .  .   8  are  supposed  to  be  equally  well  measured. 

(i)   The  Angle  Equations. 

The  angle  equations  from  the  triangles  SCW,  CWK,  IVKS, 
may  be  written  in  general  terms 

?'l  +  57,  +  7-3  +  7-4  =  /j, 
7'3  +  ^'4  +  ^'5  +  ^'6  =  h, 
7'5  +   ^6  +    ^7  +  ^8    =   h- 

As    these    equations    are    entangled,    if   we 

adjusted  for  each  in  succession  a  great  many 

repetitions  of  the  adjustment  would  be  neces- 

'^■^^'  sary  to  obtain  values   that   would   satisfy  the 

equations    simultaneously.      It    is,  therefore,    better   to    adjust 

simultaneously,  and  it  happens  that  a  very  simple  rule  for  doing 

this  can  be  found. 

Call  C^,  C,,  C,  C^,  the  correlates  of  the  equations  in  order ; 
then  the  correlate  equations  are 


<^i              =  ^'v 

a+  Q  =  v„ 

c;        =  v^_, 

c  +  c;  =  v„ 

C^-\-  Q  =  vs, 

Cs  =  v^, 

c,+  a  =  v„ 

Cz  =  J'8, 

and  the  normal  equations. 


4  Ci  +  2  C,  =  /p 

2   Cj   +    4  C,  +   2   Cg  =  /,, 

2     C     +     4    C3     =     /g. 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES     229 
Solving  these  equations,  there  result, 

<^i  =  H+  3  A  -  2  /,  +     /s), 

C3=i(-    2/^   +   4/,-    2/,), 

Q=H+    A ---/,  + 3 4). 

Substitute  these  values  in  the  correlate  equations,  and 
^'i  =  '^2  =  i  (+  3A  -  2/2+     4), 

7^3  =  ?'4=i(-  A  +  2/3-  /,). 
^'5  =  7'6  =  i(-  A  +  2/3+  4), 
77=^'8=H+        ^1-2/2  +   3/3), 

which  may  be  written, 

^'i  =  7^2  =  i/i-H4- 1/3-^1), 

^5  =  ?'6  =  l4  +  H4-i4-^/i), 

whence  follows  at  once  the  convenient  rule  for  adjusting  the 
quadrilateral,  so  far  as  the  angle  equations  are  concerned: 

(a)  Write  the  measured  angles  hi  order  of  azimuth  in  two 
sets  of  four  each,  the  first  set  being  the  angles  of  SCW,  and  the 
second  those  of   WKS. 

(/S)  Adjust  the  angles  of  each  set  by  oncfourth  of  the  dif- 
ference of  this  sum  from  180°  +  excess  of  triangle,  arranging 
the  adjusted  angles  in  two  columns,  so  that  the  first  column  will 
show  the  angles  of  SCK,  and  the  second  those  of  CWK. 

(7)  Adjust  the  first  column  by  one-fourth  of  the  difference  of 
its  stmt  from  180°  4-  excess  of  triangle,  and  apply  the  same  cor- 
rection, with  the  sign  changed,  to  the  second  column. 

This  rule  was  first  published  by  me  in  the  Journal  of  the 
Frankli)i  Institute,  June,  1880. 

The  spherical  excesses  of  the  trianj>;lcs  .S'C^F,  CII'A'.  II'A'S,  heinj^  o.  16", 
0.35",  and  0.47",  respectively,  the  adjustment  of  the  quadriiatcial  may  be 
arranged  as  follows : 


230 


THE  ADJUSTMENT  OF  OBSERVATIONS 


Measured  Angles. 


33 
62 

58 

25 

53  35- 
03  27 

44  38 
18  16 

14 
56 
98 

80 

i8o  +  e  = 

179 

180 

59  58 
00  00 

48 
16 

58 
37 

27 

56 

4)1 

68 

0 

54  57 
02  04 

38  46 

24    Q 

42 

54 
•43 
48 

77 

i8o  +  e  = 

179 
I  So 

59  58 
00  00 

.22 

•47 

4)2 

•2S 

0 

•56 

35-56 
27.98 


Adjusted  Angles. 


39-40 
17.22 


58.11 
04.99 


47.04 
10 -33 
00.91 
00  .28 

4)0-63 
0.16 


35-40 
27  .82 
39-56 
17-38 
00.16  check 


58-27 

05-15 
46.88 
10.17 
00.47  check 


169.  (2)     T/ic  Side  Equation. 

Using  the  values  of  the  angles  just  found,  we  next  form  the 
side  equation  with  pole  at  O.      It  is 

sin  OSC  sin  OCW  sin  OWK  sin  OKS  _ 
sin  SCO  sin  CWO   sin  WKO   sin  KSO  ~  ^' 

or  writing  it  in  general  terms,  when  reduced  to  the  linear  form, 

a^vl  +  rt-,?'/  +  a^v.^  +  a^vl  +  a^^'^  +  a^v^  +  a,{i<^  +  a^v^  =  1  ^, 

where  -z'/,  v',  .  .  .  are  the  corrections  resulting  from  the  side 
equation. 

Solving  as  in  Ex.  2,  Art.  120,  we  have  the  corrections 

/  _  Jh/         r  _     ^2     7 

1>.  f  ^   li,      Z'2         r-  -1    '4)       '       '       * 

L^r<'?]  [aaj 

These  corrections  may  be  found  still  more  rapidly  as  follows : 
Since  the  side  equation  may  be  so  transformed  that  the  coeffi- 
cients «j,  a.„  .  .  .  are  approximately  equal  to  unity  numerically 


ADJUSTMENT    OF    A    TRIANGULATION  — ANGLES       23 1 

(see  Art.    149),  we  may  take  each  of   them  to  be  unity,  and 
then 

^1     —    ^3     —    ^5     —    '7     —  T^    8    '4> 

,,/    7,   /     71/     r,  '    1.    /    • 

that  is,  the  corrections  to  the  angles  are  numerically  equal,  but 
are  alternately  +  and  — . 

This  plan  has  the  additional  advantage  of  not  disturbing  the 
angle  equations.  The  rule  gives  approximate  results  which  are 
the  more  nearly  correct,  the  more  nearly  the  coefficients  a^,  a.„ 
.  .  .  are  equal  to  each  other  and  the  smaller  is  the  absolute 
term  of  the  side  equation.  In  many  cases  it  will  give  results 
which  depart  widely  from  those  found  by  the  exact  process. 

Returning  to  our  numerical  example,  we  first  reduce  the  side  equation  to 
the  linear  form 

oil)  Tt  t  II 

OSC     =  ij,  53  35.40  +  7/,,  SCO     =  62  03  27.82  +  7/2, 

OCW  =  58  44  3956  +  7/„  CWO  =  25  18  17.38  +  ^4, 

OWK  =  58  54  58.27  +  v^,  IV KO  =  37  02  05.15  +  Vf„ 

OKS   =  27  38  46.88  +  v„  KSO    =  56  24  10.17  +  T/g. 

9-7463587  +   31  -3  "^^1  9-9461673  +   11.2  7/, 

9.9318952  +   12.87/3  9.6308691   +44.57/4 

9.9326832  +   12.77/5  9.7797125   +   27.97/0 

9.6665301   +  40.27/7  9.9206181   +   14.07/g 

72  70 

70 
2 

Dividing  by  20,  which  will  reduce  the  coefficients  to  unity  appro.ximately, 
and 

1 .  56  7/,'  —  0.56  7/0'  +  o .  64  7/3'  —2.22  7/4'  +  o .  64  7/5' 

—    I  .  40  7/('  +2.01  7/7'  —  O  .  70  7/>j'  +0.10=0. 

Hence,  [aa]  =  15, 

and  7/,'  =—  o.oi",     z'/  =  0.00",     7//  =  0.00",     7/.,'  =+  o.oi",  etc. 

By  the  second  rule  the  corrections  would  be  ^  -77— >  that  is,  ^  0.01",  alter- 

o 

nately,  which  values  differ  but  little  from  the  preceding. 

The  total  corrections  to  the  angles  are  the  sums  of  the  two  sets  of  cor- 
rections from  the  angle  and  side  equations. 

170.  (a  2)  Adjustment  of  a  Quadrilateral :  Rigorous  Method. 

—  By  the  fol](jwing  artifice  the  quadrilateral  may  be  rigorously 


^32       THE  ADJUSTMENT  OF  OBSERVATIONS 

adjusted  for  the  side  equation  without  disturbing  the  angle 
equation  adjustment,  which  amounts  to  the  same  thing  as  the 
simultaneous  adjustment  of  the  angle  and  side  equations. 

Suppose  that  the  angle  equations  have  been  adjusted  as  al- 
ready explained  in  {a  i).  If  v^,  <,  .  .  .  v^,  denote  the  cor- 
rections arising  from  the  side  equation,  the  condition  equations 
may  be  written 

■vl  +  v^  +  V^  +  Vi   =  o, 

•Vz  +  vi  -h  v^  +  vl  =  o, 

-vl  +  vl  +  V^  +  Z'/  =  o, 
a.pl  +  a-pl  +  flags'  +  «4^'4'  +  ''5^5'  +  «6V  +  a-i^n  +  ^s'^^s'  =  K- 

By  writing  the  corrections  in  the  form 

v^  =  +  v  -  v',  v^  =  +  v  —  v", 

v^  =  -v-\-  v",  v-l  =  -v  +  v"", 

vl  =  —  v  -  v",  v^   =-v  —  V      , 

the  first*  three   condition   equations  become   0=0  identically, 

and  we  have  therefore  to  deal  only  with  the  single  condition 

equation 

((?!  +  a,  +  (75  +  Oe  -  fls  -  <?4  -  c^  -  ^s)  ^'  +  (<?i  -  '^)  ^'' 

with 

(v  +  vj  +  {v-  vj  +  {-v  +  v'J  +  {-v-  v'J  +  .  .  .  =  a  min. 
The  correlate  equations  are, 

((7i  -f  a,  +  '^s  +  t76  —  (73  —  «<  —  t^7  —  ^'s)  C  =  4  'Z'. 

((7i  —   d,)    C  =       1;', 
(dg    —    (74)    C   =        V", 

(05  -  a^  C  =  z'"', 
((77  —  a^  C  =  v"". 
Substitute  in  the  condition  equation,  and 

C   {\  ((7i   +  >\  +  Cr,+  a^-   (73  -  (74  -   (77  "  ^s)"  +   0?1  "   03)^ 

+  ((73  —  (74)^  +  ('^5  —  "e)^  +  (('7  -  a^f)  =  ^4' 
from  which  C  can  be  found. 


ADJUSTMEXT    OF    A    TRIANGULATIOX —ANGLES       233 
Hence  the  corrections  are  known. 

The  complete  adjustment  of  our  quadrilateral  is  contained  in  the  foUow- 
ins:  table  : 


Meas.  Angles. 

Local 

Log 

Sines. 

Log  DiFF. 

Sqs. 

w 

0 

33     S3 
62     03 
58     44 
25     18 

35  '4 
27.56 
3»-98 
16.80 

58.48 
00.16 

1.68 

35-56 

27.98 

3940 
17.22 

35-4° 
27.82 
39-56 
17-38 

9  7463587 
9.9318952 

9.9461673 
9.6308691 

+  31.3 

+  II. 2 
+  12.8 

+  44-5 

42.5 

57-3 

1806 
3283 

58     54 
37     02 
27     38 
56     24 

57-54 
0443 
46.48 
09.77 

58.11 

04.99 
47.04 

IO-33 

58.27 
05- « 5 
46. 88 
10.17 

9.9326832 
9.6665301 

9.7798123 
9.9206181 

+  .2.7 

+  27.9 
+  40.2 

+  M-o 

40.6 
54.2 

164S 
2938 

58.22 

00.91 

— 

— 

00.47 

00.28 

72 

70 

102.5            92'I 

2.25 

0.63 

70 

2 

92.1 

4)10.04 

2.6 

5-2 

27 

9702 

V 
2^ 


V 
42-5 


57-3       40-6 


54- 


9702 


Hence  the  corrections  are  known.     These  corrections,  applied  to  the  local 
angles,  give  the  final  angles  required- 

171.    {b)  Adjustment  of  a  Single  Triangle. 
Rule  and  example  in  E.x.  2,  Art.  118,  and  Ex.  2,  Art.  120. 
The  sin<;le  triangles  in  our  figure  are  Kingston,  Wolfe,  Am- 
herst ;  Wolfe,  Duck,  Amherst  ;  Vanderlip,  Oswego,  Sodus. 

For  example,  take  the  first  (e  =  0.72"): 


I 

ilEASURnn. 

Adjusted. 

Kingston     . 

88 

19      14.70 

15-78 

Wolfe     .      .      . 

.S.S 

59      20.ro 

21  .18 

Amherst 

35 

41      22.69 

23-77 

179 

59     57-49 

0.73  check 

180  +  e  = 

180 

00     00.72 

3)3-23 

1.08 

234 


THE    ADJUSTMENT    OF    OBSERVATIONS 


172.  (r)  Adjustment  of  a  Central  Polygon.  —  In  the  cen- 
tral polygon  Duck,  Amherst,  Grenadier,  Stony  Point,  Oswego, 
Vanderlip,  the  condition  equations  in  general  terms  are  : 


Local  equation  (horizon  equation), 

■^3  +   ^6   +  ^^'9  +  ^'l2  +  ^15  =   ^1- 

Angle  equations, 

^1  +  ^2  +  ^'3  =  h^ 

Vi    +    Vr,    +    ^^6    =    h, 


t\3  +   "^'14  +  ^15 


Side  equation  (pole  at  Duck), 


+  ihi^ii  =  I' 


We  may  adjust  for  these  equations  in  order,  first  the  horizon 
equation,  then  the  angle  equations  separately,  as  they  are  not 
entangled,  and  next  the  side  equation. 

A  rigorous  adjustment  may,  however,  be  carried  out  at  once 
with  very  little  additional  labor.  Adjust  first  each  angle  equa- 
tion by  itself,  and  let  (7' J,  {v,),  ...  be  the  values  that  result. 
Let  (i),  (2),  .  .  .  denote  the  farther  corrections  to  the  measured 
angles  in  order  arising  from  the  local  and  side  equations,  so  that 

V,  =  (%\)  +  (i), 

^2  =  (^2)   +  (2)» 


If  we  substitute  these  values  in  the  above  equations  we  have 
the  new  condition  equations, 

a,(i)  +  aA2)  +  •  •  •  +r„(i4)  =l\ 
(3)  +(6)+  (9)  +  (12)  +  (15)=/". 
(i)  +    (2)  +    (3)  =  o'\ 


ADJUSTMENT    OF    A    TRIANGULATION —  ANGLES      235 
(4)+    (5)+     (6)  =  o, 
(13)  +  (U)  +  (15)  =0, 

from  which  to  find  (i),  (2),  (3),  .  .  .  (i5)- 

Calling   Cj,   C  I,  II,   .   .   ■   the  correlates    of    the    condition 
equatiojis  in  order,  we  have  the  correlate  equations 

(i)  =  a,C,  +  /,  (4)  =  ^'A  +  11  .... 
(2)  =  a.C,  +  /,  (5)  =  a,C,  +  11  .... 
(3)=     C,  +  I,         (6)=     C  +  //         .... 

Eliminate  now  the  angle  equation  correlates.     By  addition, 

(a,  +  a;)C,  +  3l    +C,  =  o, 
(cu  +  a,)  c,  +  3ri  +  c,  =  o, 


Hence 


(i)  =+  1  (2<7^-    a,)Q-ia, 
(2)  =-i(    ./, -2a,)Q-iC, 

(3)=-H  ''1+  'OQ  +  ta., 


Substitute  these  values  of  (i),  (2),  .  .  .  in  the  condition  equa- 
tions, and  we  have  the  normal  equations 

2  {[</<;]  -  ^1(7.,  -  (-4  .-5  -  •  •    }  C\-  [a]  C  =  3  /', 
-[u]Q-hioC,  =  3/". 

Solving,  we  find  C,,  C,,  and  thence  (i),  (2),  .  .  .  are  known. 

The  normal  equations  may  be  written  down  directly  in  every 
case,  as  the  law  of  their  formation  is  as  evident  as  that  of  ordi- 
nary normal  equations.  They  involve  only  two  unknowns,  ( ^, 
C.^,  no  matter  how  many  sides  the  polygon  has. 

173.    We  shall  now  proceed  with  our  numerical  example. 

At  station  Duck  the  measured  values  of  the  angles  are  taken, 
at  the  other  stations  the  locally  adjusted  angles. 


>36 


THE    ADJUSTMENT    OF    OBSERVATIONS 


CliVEN  Angles. 

Log  Sines. 

DiF.  l". 

Squares. 

Prod- 
ucts. 

Sums. 

0    /     // 

// 

54  38  00.34 

00.62 

9.91 14060 

+  14.9 

222.0 

257-8 

-  2.4 

50  35  04.19 
74  46  55-83 

04.47 
56.12 

9-8879338 

+  17-3 

299-3 

180  00  00.36 

iSo  00  01.21  = 

=  i8o  +  e 

3)0.85 

0.28 

78  13  33.84 
51  53  12.70 

34-49 
13-35 

9.9907654 

9.895S618 

+  4.4 
+  16.5 

19.4 
272.2 

72.6 

—  12. 1 

4953  12.77 

13-42 

59-31 
1.26 

1-95 
0.65 

88  22  00.86 

00.47 

9-9998235 

+  0.6 

•4 

21. 1 

-34-6 

30  53  44.32 
60  44  iq.46 

43-03 
19.07 

9.7105188 

+  35-2 

1239.0 

04.64 

3-47 

1. 17 

0-39 

26  49  17.37 
40  01  45.86 

18.  IS 
46.64 

9-6543842 

9-8779750 

+  41-6 
+  18.3 

1730.6 
334-9 

761.3 

+  23-3 

104  08  58.93 

59-70 

2.16 

4-49 

2-33 
0.78 

38  18  07.3a 
71  15  25-32 

06.33 
24.36 

9-7922537 

9-9763353 

+  26.7 
+  7-1 

712.9 
50-4 

189.6 

+  19.6 

70  26  31.99 
04.61 

31.02 

632S      6247 

4881. I 

1302.4 

-  6.2 

1. 71 

6247 

1302.4 

2.90 

81 

6183.5 

0.97 

3 

2 

24  1 

I  2,367.0 

The  Nor))ial  Equations. 


Local  Equation  at  Station  Duck. 


12,367  C,  +  6.2  Co  =  -243 

74 

46 

56-12 

6.2  Ci  4-10   C2  =    2.01 

49 

53 

13-42 

60 

44 

19.07 

.*.  C,  =  —  0.020 

104 

08 

59-70 

C2  =  +  0.213 

70 

26 

31.02 

359 

59 

59-33 

360 

00 

00.00 

00.67 
3 


ADJUSTMENT    OF    A    TRIANGULATION  —  ANGLES       237 


c 

ORRECTIONS. 

Adjusted 

0         / 

Anc 

LES 

(i: 

=  - 

0-39 

54 

38 

00.23 

(2: 

=  + 

0.26 

50 

35 

04 

73 

(3 

=   + 

0.13 

74 

46 

56 

25 

(4; 

=  - 

0.24 

78 

13 

34 

25 

(5; 

=  4- 

0.18 

51 

53 

13 

53 

(6^ 

=  + 

0.06 

49 

53 

13 

48 

(7; 

=  - 

0.31 

88 

22 

00 

16 

(8^ 

=  4- 

0.40 

30 

53 

44 

7,2, 

(9) 

=  - 

0.09 

60 

44 

18 

98 

(10: 

=  — 

0-75 

26 

49 

17 

40 

(11] 

=  4- 

0.45 

49 

01 

47 

09 

(12) 

=  4- 

0.30 

104 

08 

60 

00 

(13) 

=  - 

0.47 

38 

18 

05 

86 

(14) 

=  4- 

0 .  20 

71 

15 

24 

56 

(15) 

=  + 

0 .  27 

70 

26 

31 

29 

Approximate  Method  of  Finding  tJie  Precision. 

174.  An  adjustment  may  be  carried  out  rigorously  so  far  as 
finding  the  values  of  the  unknowns  is  concerned,  but  only  an 
approximate  value  of  the  m.  s.  e.  of  the  angles  or  sides  may  be 
thought  necessary. 

In  good  work  the  following  method  will  give  results  nearly 
the  same  as  those  found  by  the  rigorous  process. 

The  average  value  /x'  of  the  m.  s.  e.  of  an  angle  in  a  triangu- 
lation  net  after  adjustment  is  easily  seen  from  Art.  1 13  to  be 


=v/^ 


Mm. 


where 


n   =  number  of  angles  observed, 

n^  =  number  of  local  and  general  conditions, 

/x,„=  m.  s.  e.  of  a  measured  angle  of  average  weight ; 


or,  if  all  the  angles  are  of  equal  weight,  it  is  the  m.  s.  c.  of  a 
measured  angle. 

The  value  of  /i,  the  m.  s.  e.  of  an  angle  of  unit  weight,  is,  by 
the  usual  formula, 


238  THE    ADJUSTMENT    OF    OBSERVATIONS 


-/-fi- 


u, 

in  which  /,„  is  the  average  weight  of  a  measured  angle. 

To  find  the  m.  s.  e.  of  a  side  of  a  triangle,  a  single  chain  of 
the  best-shaped  triangles  between  the  base  and  the  side  is 
selected,  all  tie  lines  being  rejected.  Then,  assuming  the  base 
to  be  exact  and  the  m.  s.  e.  of  each  adjusted  angle  to  be  ^l',  we 
have  from  Ex.  9,  Art.  126, 

fj^^iog  «n  =  f  /A  ^  \}/  +  8jr  +  8a8b], 

where  S„  8^  are  the  log  differences  corresponding  to  i"  for  the 
angles  A,  B  in  3.  table  of  log  sines. 

Ex.  — To  find  the  m.  s.  e.  of  the  side  OL  as  derived  from  the  base  A^S 
in  the  figure  OA^SL  (Fig.  19). 

Number  of  angles  measured  =  9. 

Number  of  conditions,  local  and  general,  =  5. 

From  the  adjustment  (Art.  157)  [pv^]  =  7.54. 


.-./t  = 


=  1-23 

1.23 
/>m  =  10.4  and  .-.  f^m  =     ,  =  0.38 


10.4 


CHAPTER  VII 

APPLICATION    TO    THE    ADJUSTMENT    OF    A    TRIANGULATION. 
METHOD    OF    DIRECTIONS 

175.  This  method  is  due  to  Bessel.  Various  modifications 
of  Bessel's  plan  of  making  the  observations  are  in  use  on  differ- 
ent surveys.  The  following  is  that  used  in  the  Coast  and  Geo- 
detic Survey  on  primary  triangulation  at  the  present  time. 

Each  series  of  observations  consists  of  successive  pointings 
on  the  various  stations  in  order,  from  left  to  right,  with  corre- 
sponding readings  of  the  horizontal  circle  with  three  micrometer 
microscopes,  followed  immediately  by  pointings  on  the  same 
stations  in  the  reverse  order,  after  reversing  the  position  of  the 
horizontal  axis  of  the  telescope  in  the  wyes,  and  turning  the 
alidade  180°  in  azimuth,  each  pivot  remaining  in  contact  with 
the  same  wye  as  before.  Each  observation  of  an  angle  consists 
therefore  of  two  pointings  on  each  station  involved,  one  in  each 
position  of  the  telescope,  together  with  the  corresponding  microm- 
eter readings,  twenty-four  in  all,  both  a  forward  and  a  back- 
ward reading  of  each  micrometer  being  made  in  each  of  its 
positions.  Sixteen  such  series  of  observations  are  taken  upon 
each  station,  one  in  each  -of  sixteen  positions  of  the  horizontal 
circle. 

As  implied  in  this  statement  of  the  method,  the  instrument 
used  carries  an  accurately  divided  horizontal  circle  which  is 
read  by  micrometer  microscopes.  The  circle  may  be  shifted  to 
different  positions  in  azimuth,  but  is  not  provided  with  such  a 
clamp  and  slow  motion  tangent  screw  controlling  the  position 
of  the  graduated  circle  a»s  is  needed  on  any  instrument  used 
with  the  method  of  repetitions. 

In  the  repetition  method  of  angles,  one  angle,  between  two 

239 


240  THE    ADJUSTMENT    OF    OBSERVATIONS 

Stations,  is  measured  at  a  time.  In  the  direction  method,  as 
practiced  in  the  Coast  and  Geodetic  Survey,  a  single  series  oi 
observations  serves  to  measure  all  the  angles  between  stations, 
or  to  determine  the  relative  directions  to  all  stations,  observed 
upon  in  that  series.  In  the  repetition  method,  the  unknowns 
which  are  being  measured,  are  the  angles  ;  and,  in  the  direction 
method  described,  the  unknowns  are  relative  directions,  and  the 
difference  of  any  two  such  unknowns  is  an  angle.  As  this  dif- 
ference exists  in  the  method  of  observation,  it  also  exists  in  the 
method  of  adjustment.  In  the  direction  method  of  adjustment, 
the  unknowns  are  directions,  one  for  each  line  observed  over 
from  each  station,  and  angles  appear  only  as  differences  of 
directions. 

176.  In  the  direction  method  of  adjustment,  it  is  assumed 
that  there  is  an  error  inherent  in  each  direction  observed  which 
affects  every  angle  involving  that  direction.  For  errors  which 
are  due  to  the  instrument,  and  also  those  due  to  the  observer 
(with  one  exception,  noted  below),  there  is  no  sufficient  reason 
for  assuming  that  errors  are  inherent  in  the  directions,  rather 
than  in  the  angles.  All  errors  due  to  external  conditions,  that 
is,  to  conditions  outside  the  instrument  and  the  observer,  on  the 
other  hand,  must  be  assumed  to  be  inherent  in  the  separate 
directions  observed,  rather  than  in  the  angles.  The  external 
errors  may  be  due  to  various  causes,  including  phase  and  asym- 
metry of  the  object  pointed  upon,  eccentricity  of  the  signals 
pointed  upon,  or  of  the  instrument,  and  lateral  refraction. 
Such  errors  tend  to  recur  with  one  sign  for  each  observation 
over  a  given  line  from  a  station,  and  therefore  to  affect  that 
direction  in  a  constant  manner  which  is  independent  of  the 
other  directions  which  happen  to  enter  that  series  of  observa- 
tions. In  this  class  of  external,  constant  errors  must  also  be 
placed,  whatever  tendency  the  observer  may  have  to  misjudge 
the  position  of  the  center  of  the  image  pointed  upon.  If  an 
observer  makes  every  pointing  on  every  object  too  far  to  one 
side,  say  to  the  left,  by  a  constant  amount,  this  error  will  not 


ADjUST;.IEXT    of    a    TRIAXGULATION  — directions     241 

affect  the  final  results.  But  any  such  tendency  to  a  one-sided 
pointing  is  reasonably  certain  to  be  modified  by  the  brightness 
and  size,  as  well  as  by  any  asymi"  ^try,  of  the  image  pointed 
upon.  It  is  therefore  probable  that  an  error  of  bisection  exists 
which  is  peculiar  to  each  direction  observed,  as  long  as  the 
image  of  the  object  pointed  upon  in  that  direction  remains  con- 
stant in  appearance. 

It  matters  little  in  choosing  the  method  of  adjustment, 
whether  the  above  outline  of  the  manner  in  which  errors  are 
inherent  in  particular  directions  is  accurate  in  detail  or  not.  If 
the  great  mass  of  evidence  available  indicates  strongly  that  the 
principal  errors  in  angle  measurements  are  of  the  external  class, 
and  that  they  are  inherent  in  separate  directions  rather  than  in 
angles,  the  direction  method  of  adjustment  should  be  used  in 
preference  to  the  angle  method.  That  the  direction  method 
should  be  used,  and  for  this  reason,  is  the  conviction  to  which 
years  of  critical  observation  have  led  the  computers  of  the  Coast 
and  Geodetic  Surv^ey.  The  direction  method  of  adjustment,  as 
set  forth  in  this  chapter,  is  now  almost  exclusively  used  in  that 
Survey,  even  when  the  observations  have  been  taken  by  the 
method  of  repetitions. 

177.  In  the  first  part  of  the  preceding  section  the  direction 
method  of  observation  in  use  in  the  Coast  and  Geodetic  Survey 
is  described.  The  U.  S.  Lake  Survey  is  at  present  using  a 
method  of  observation  with  a  direction  instrument  which  differs 
widely  in  several  important  respects  from  the  Coast  and  Geo- 
detic Survey  method.  The  most  important  difference  is  that 
the  different  angles  are  measured  independently,  that  is,  there 
are  but  two  signals  pointed  upon  in  each  series  of  observations. 

The  advocates  of  the  method  of  independent  angles  with  a 
direction  instrument  urge  (i)  that  if  the  twist,  due  to  the  action 
of  the  sun's  rays,  of  the  tower  upon  which  the  instrument  is 
mounted,  be  considered;  (2)  and  if  the  influence  on  distinctness 
of  vision  of  the  use  of  the  same  focus  for  lines  of  different 
lengths;   (3J   the  interruptions  that  may  occur  in  the  course  of 


2  42  THE    ADJUSTMENT    OF    OBSERVATIONS 

a  long  series  ;  and  (4)  the  more  uniform  line  that  may  always 
be  had  when  the  number  of  signals  in  use  at  any  one  time  be 
small,  —  be  also  considered,  the  conclusion  must  be  reached  that 
this  method  will  give  a  greater  accuracy  than  the  other. 

The  advocates  of  the  method  of  including  all  the  signals 
which  are  then  visible  in  each  series,  believe  that  the  second 
and  fourth  considerations  are  of  minor  importance  under  actual 
average  conditions ;  that  the  third  consideration  is  of  little 
importance  if  one  does  not  wait  (and  he  should  not)  for  a  signal 
which  is  not  showing  when  a  pointing  is  desired ;  and  that  on 
the  towers,  as  now  built,  in  the  Coast  and  Geodetic  Survey, 
the  twist  is  so  small  as  to  be  difficult  to  discover,  even  by 
special  observations  for  that  purpose,  extending  over  long  inter- 
vals of  time,  and  therefore  that  the  first  consideration  is  of  little 
importance.  As  against  the  considerations  set  forth  above 
bearing  upon  the  accuracy  to  be  obtained,  these  advocates  call 
attention  to  the  fact  that  in  their  method  the  pointings  upon 
each  signal  are  scattered  over  the  whole  of  the  observing  period 
during  which  that  signal  is  visible  ;  whereas,  in  the  method  of 
independent  angles,  the  observations  on  each  signal  are  confined 
within  a  few  short  periods  ;  and  that,  therefore,  a  greater  variety 
of  conditions  are  encountered,  and  a  tendency  to  greater  accu- 
racy secured  in  the  former  method.  The  great  disadvantage  of 
the  method  of  independent  angles  lies  in  the  greater  time  and 
cost  required  to  secure  a  given  number  of  observations.  As 
compared  with  the  method  of  independent  angles,  the  method 
used  in  the  Coast  and  Geodetic  Survey  requires  but  three- 
fourths  as  many  pointings  for  a  given  number  of  observations 
of  each  angle  if  three  signals  are  observed  in  each  series  on  an 
average,  and  but  five-eighths  as  many  if  there  are  five  observed 
in  each  series. 

It  has  been  urged  in  favor  of  the  method  of  independent 
angles  that  it  simpHfies  the  local  adjustment.  This  argument 
had  considerable  force  as  against  the  method  formerly  used  in 
the  Coast  and  Geodetic   Survey,  but  not  against  the  present 


ADJUSTMENT    OF   A   TRIANGULATION  — DIRECTIONS     243 

practice  in  that  Survey,  with  which  no  local  adjustment  what- 
ever is  found  to  be  necessary. 

178.  The  Local  Adjustment.  —  If  all  signals  are  observed 
in  every  series,  the  local  adjustment  becomes  simply  a  process 
of  taking  means  and  differences.  If  a  broken  series  is  observed, 
that  is,  a  series  in  which  one  or  more  of  the  signals  are  missing, 
because  they  were  not  visible  at  the  particular  time  when 
needed,  the  Coast  and  Geodetic  Survey  observers  are  at  present 
directed  that,  "the  missing  signals  are  to  be  observed  later  in 
connection  w^ith  the  chosen  initial,  or  some  other  one,  and  only 
one,  of  the  stations  already  observed  in  that  series."  The 
observations  on  the  signal  which  is  common  to  the  two  frag- 
ments of  a  series  are  used  to  connect  these  fragments,  and  when 
so  connected  the  series  is  used  as  if  it  had  all  been  observed  at 
one  time,  and  no  local  adjustment  is  necessary.  In  this  process 
the  series  which  arc  made  up  of  joined  fragments,  are  given  a 
slightly  greater  weight  than  they  should  be.  But  this  is  of  very 
little  importance,  especially  as  the  principal  errors  in  angle 
measurements  are  of  the  systematic  class  and  do  not  appear 
until  a  figure  adjustment  is  made. 

If  the  observations  are  made  by  the  method  of  independent 
angles  with  a  direction  instrument,  as 
in  the  Lake  Survey,  the  local  adjust- 
ment may  be  made  by  the  methods 
stated  in  articles    121,    122. 

179.  Ths  Figure  Adjustment.*- 
The  following  directions  were  ob- 
served, among  others,  at  the  stations 
named.  It  is  required  to  adjust  the 
quadrilateral  shown  in  Fig.  25  by  the 

method  of  directions,  the  line  Two-  ''■s-^s- 

Rock  having  been  completely  fixed  by  the  adjustment  of  the 

preceding  figures. 

*  Throughout  this  chapter  all  formuliL-  and  examples  have  been  Riven 
upon  the  assumption  that  all  observed  directions  arc  to  be  assigned  ecjual 


244  THE    ADJUSTMENT    OF    OBSERVATIONS 

Obsen'ed  Directions. 


At  Two. 

At  Rock. 

Hill      . 

.        •       145 

3,Z 

38.1 

Two 

•      •      •       53 

30 

44-3' 

Point   . 

212 

09 

30.8 

Hill 

.      .      .       78 

36 

08.7 

Rock    . 

269 

41 

25  .2* 

Point 

.      .      .     127 

43 

40.0 

*  Corrected  direction, 

26.3 

**  Correc 

:ted  direction, 

42.7 

". 

At  Hill. 

At  Point. 

Rock    . 

•     0 

00 

00.0 

Rock 

.        .        .       165 

04 

04.8 

Two     . 

•        •         30 

46 

43-1 

Two 

.        .        .       213 

19 

10.7 

Point   . 

•        •       315 

44 

36.8 

Hill 

.        .        .       251 

41 

04.6 

The  signal  chosen  as  the  initial  happens  to  appear  in  these 
lists  in  only  one  case,  Rock  being  the  initial  at  Hill.  There  is 
no  difference  in  the  treatment  during  adjustment  of  the  initial 
station  and  any  other  station. 

The  ten  directions  for  which  it  is  proposed  to  derive  correc- 
tions are  identified  by  numbers  on  Fig.  25.  The  conv^enient 
notation  used  below  indicates  the  corrections  to  these  figures  by 
the  same  numbers  inclosed  in  a  parenthesis,  thus,  ( i )  stands  for 
the  correction  to  the  direction  numbered  i,  namely,  Point  to 
Rock.  No  corrections  are  to  be  derived  for  the  directions  Two 
to  Rock  and  Rock  to  Two,  as  these  directions  have  already  been 
fixed  by  previous  adjustment. 

The  four  condition  equations  are  as  follows,  the  three  angle 
equations  being  given  first : 

Cojtdition  Equations. 

o=-i.3-(r)  +  (2)-(8)+(io) 
o=-2.7-(5)  +  (6)  -(7)+   (9) 
o  =  -  7.1  -  (2)  +  (3)  -  (4)  +    (6)  -  (7)  +  (8) 
o  ==  -  12.0  +  1.9     (i)  -  4.6  (2)  +  2.7  (3)  +  0.6  (4) 
-  3-5  (5;  +  2.9  (6)  -  4-5  (9)  +  0-6  (10) 
In  the  angle  equations  each  required  correction  to  an  angle  is 
expressed  as  a  difference  of  two  corrections  to  directions,  except 
when  one  of  the  directions  is  already  fixed,  in  which  case  the 

weight.  This  is  the  case  which  most  frequently  arises.  If  it  is  desired  to 
assign  unequal  weights.  Chapter  V  shows  how  the  weights  are  to  be  intro- 
duced. 


ADJUSTMENT    OF   A   TRIANGULATION —DIRECTIONS     245 


correction  to  the  other  direction  involved  in  the  angle  becomes 
the  same  (with  the  proper  sign)  as  the  correction  to  the  angle.* 
The  triangles  are  so  small  in  this  case  that  it  is  not  necessary 
to  take  the  spherical  excess  into  account. 

The  side  equation  has  its  pole  at  Two.  The  numerical  work 
of  forming  this  side  equation  is  shown  below  in  a  convenient 
form.  (For  an  explanation  of  the  meaning  of  the  side  equation, 
see  Art.  126.) 


Plus  Terms. 

Directions. 

Uncorrected  Angles. 

Logarithmic  Sines. 

Log  Sine,  Dif.  for 

+  10 

-2  +  3 
-5  +  6 

0         /            // 

74     12     57.3 
38     21     53.9 
30     46     43-1 

9.983308 
9 . 792860 
9 ■ 709035 

+  0.6 

+  2.7 
+  3-5 

Sum 

=  9.485203 

Minus  Te 

RMS. 

Directions. 

Uncorrected  Angles. 

Logarithmic  Sines. 

Log  Sine,  Dif.  for 

-1  +  2 

-4  +  6 

+  9 

0          /            II 
48      15      05.9 

75     02     06.3 
25     05     26.0 

9.872783 
9.985015 
9.627417 

+   1.9 
+  0.6 

+  4-5 

Sum 
Difference 

.     =  9.485215 

For  such  directions  as  occur  twice  in  the  formation,  as,  for 
example,  direction  2,  the  corresponding  coefficient  is  the  algebraic 

*  "The  first  two  angle  equations  were  purposely  selected  so  that  they 
refer  to  triangles  of  which  the  fixed  line  Two-Rock  forms  one  side.  Each 
such  angle  equation  contains  but  four  terms,  whereas  otherwise  it  would 
contain  six,  and  the  normal  equations  have  two  side  coefficients  which  are 
zero.  The  work  of  forming  and  solving  the  normal  equations  is  therefore 
reduced  by  this  selection." 


246  THE   ADJUSTMENT    OF    OBSERVATIONS 

sum  of  two  log  sine  differences  for  i",  each  taken  with  the 
sign  fixed  as  indicated  in  the  first  and  fourth  or  fifth  and  eighth 
cokimns  of  the  above  form,  together  with  the  headings,  ''phis 
terms"  and  "minus  terms."  The  log  sine  differences  for  i" 
are  given  in  units  of  sixth  decimal  place. 

This  triangulation  is  of  tertiary  character,  and  hence  the  sides 
are  computed  to  six  decimal  places  only  in  the  logarithms.  The 
correlate  equations  are  shown  below  in  the  form  indicated  in 
equations  (4)  of  Art.  iio. 

Correlate  Equations. 
-C,  +  1.9  Q=   (i) 

+  c,        -  q  -  4.6  Q  =  (2) 

+  c;  +  2.7  C,  =   (3) 
—  Q  +  0.6  Q  =    (4)    , 

-c,        -  3.5  Q  =  (5) 

+  a+Q  +  2.9  C,  =    (6) 

-C,-Q  =   (7) 

-C,  +Q  =   (8) 

+  c;         -  4.5  Q  =  (9) 
+  c;  +o.6c;=(io) 

Note  that  the  coefficients  in  the  first  column  of  these  correlate 
equations  are  the  same  as  the  coefficients  in  the  first  hne  of  the 
condition  equations,  and  that  the  second  column  and  the  second 
line  correspond,  and  so  on. 

The  normal  equations  are  shown  below,  formed  from  the 
correlate  equations  and  condition  equations  as  indicated  in 
equations  (5)  of  Art.  119. 

Normal  Equations. 

o  =  —    1.3"  +  4.0  Cj  —  2.0  C3  —    5.9  C4 

o=—    2.7"  +4     C,  +  2.0  C3  +    1.9  C4 

o  =  —   7.1"  —  2.0  Cj  +2.0  C  H-  6.0  Q  +   9.6  C4 
o  =-i2.o"  —  5.9  C^  +1.9  C,  +  9.6  Q  +73.7  C, 


ADJUSTMENT    OF    A   TRI ANGULATION — DIRECTIONS     247 

The  solution  of  these  equations  gives : 
Q  =+1.173" 

Cj  =  —  O.I  10 

C3=+i.5ii" 

C4  =  +  0.063 

These  values  substituted  in  the  correlate  equations  give  the 
required  corrections. 

(i)  =-  1.05"  (6)  =+1.58" 

.     (2)  =-0.63"  (7)  =-1.40" 

(3)  =+1.68"  (8)  =+0.34" 

(4)  =-  1.47"  (9)  =-0.39" 

(5)  =-o.ii"  (10)  =+I.2l" 

After  applying  these  corrections  to  the  directions,  the  com- 
putation of  triangle  sides  shows  every  triangle  with  the  sum  of 
its  corrected  angles  equal  to  180°  00'  00.0",  there  being  no 
spherical  excess  in  these  small  triangles,  and  shows  the  lengths 
of  the  lines  to  be  the  same  when  computed  in  two  possible  ways 
through  the  triangles.  The  correctness  of  the  adjustment  is 
thus  checked. 

The  procedure  in  adjusting  an)'  figure,  however  complicated, 
in  which  the  only  thing  fixed  by  previous  adjustment  is  one  line, 
is  not  essentially  different  in  any  respect  from  that  here  illus- 
trated by  the  simple  case  of  a  quadrilateral  with  one  fixed  line. 
The  number  of  angle  and  side  condition  equations  in  any 
particular  figure  is  to  be  determined  as  indicated  in  Arts,  144, 
151,  152.  For  examples  of  condition  equations  for  complicated 
figures,  see  Appendix  4  of  the  Coast  and  Geodetic  Survey 
Report  for  1903. 

180.  The  Best  Side  Equations.  —  In  Arts.  153,  154,  certain 
suggestions  arc  given  as  to  the  manner  in  which  side  equations 
should  be  selected  to  avoid  the  danger  that  the  solution  of  the 
normal  equations  may  be  an  unstable  one,  that  is,  a  solution  in 
which  the  effect  of  omitted  decimal  places  on  the  derived  values 
of  the  required  unknowns  is  large,  and  in  which  it  is,  therefore, 


248  THE    ADJUSTMENT    OF    OBSERVATIONS 

necessary  to  carry  a  large  number  of  decimal  places  in  the 
solution  to  secure  the  unknowns  with  certainty  to  a  small 
number  of  decimal  places.  The  suggestions  are  difficult,  but 
important,  to  follow.  This  article  will  serve  to  illustrate  these 
suggestions  by  a  concrete  case,*  namely,  that  shown  in  Fig.  25, 
The  observed  directions  are  as  follows: 

At  Spear.  At  Topacco  Row. 

O  t  I'  O  I  II 

Long     .     .     .  o  00  00.000  Spear    ...       o  00  00.000 

Smith    .      .      .  6  04  57.749  Long      ...      72  37  08.593 

Flat  Top    .      .  37  00  48.900  Smith    .      .      .118  11  11. 341 

Tobacco  Row,  47  03  16.925  Flat  Top    .      .    159  40  31.200 

At  Long. 

o 

Smith o  00  00.000 

Flat  Top 57  52  28.128 

Tobacco  Row     .      .      .  108  47  15.636 

Spear 169  06  53.169 

At  Smith.  At  Flat  Top. 

o         )  o  I  n 

Flat  Top    .     .     o  00  00.000  Tobacco  Row,  o  00  00.000 

Tobacco  Row,  30  12  41.103  Spear    ...  10  17  00.258 

Spear    .      .      .   51  03  16. 151  Long     .     .      .  42  01  51.794 

Long     .      .      .  55  51  27.879  Smith    .      .      .  108  18  02.385 


The  figure  requires  three  side  equations.  Let  each  side 
equation  be  represented  symbolically  by  the  abbreviation  inclosed 
in  a  parenthesis  for  the  station  used  as  a  pole,  followed  in  order 
by  the  abbreviations  for  each  of  the  other  stations  at  the  ends 
of  the  sides  involved.  As  in  Art.  179,  the  required  correction 
to  a  direction  will  be  indicated  by  the  number  of  that  direction 
inclosed  in  a  parenthesis.  The  numbers  assigned  to  the  direc- 
tions are  indicated  on  the  figure.  The  side  equations  repre- 
sented by  the  symbols  (L.)-Sm.-T.  R.-Sp.,  (L.)-Sm.-F.  T.-Sp., 
and  (L.)-Sm.-F.  T.-T.  R.  are 

*  Thi.s  article,  as  well  as  the  suggestions  in  Art.  154,  is  based  on  pp.  118- 
120  of  the  C.  and  G.  S.  Rep.  for  1878  (App.  No.  8)  written  by  iMr.  M.  H. 
Doolittle. 


ADJUSTMENT   OF    A   TRIANGULATION  — DIRECTIONS    249 

o  =  -h  1.2"  -  17.80  (i)  +  19.76  (2)  -  1.96  (4)  -  0.66  (5) 
+  2.72  (6)-  2.06  (7)  -  4-38  (14)  +  25.05  (15) 

—  20.67  (16).  (0 
o  =  -  1.9"  -  16.97  (i)  +  19.76  (2)  -  2.79  (3)  -  1.43  (13) 

+  25.05  (15)  -  -3-62  (16)  -  3.40  (18)  +  4.33  (19) 

—  0.93  (20).  (2) 
o  =-  1.9"  -  1.95(6)  4-  2.06(7)  -0.1 1  (8)  -  1.43(13) 

+  4.38  (14)  -  2.95  (16)  -  2.33  (17)  +  3.26  (19) 

—  0.93  (20).  (3) 

The  directions  i,  2,  15,  and  16  are  the  sides  of  the  small 
angles  at  Spear  and  Smith.  In  the  first  two  equations  the 
coefficients  of  the  corrections  (i),  (2),  (15),  and  (16)  so  largely 
predominate  over  everything  else,  and  for  corresponding  terms 
are  so  nearly  equal,  that  the  two  equations  may  be  considered 
approximately  identical.  The  effect  of  this  would  be  to  make 
certain  side  coefficients  in  the  normal  equations  about  as  large 
as  the  corresponding  diagonal  coefficients,  and  the  solution  would 
be  unstable.  An  attempt  at  solution  with  factors  extending  to 
but  three  significant  figures  would  not  be  likely  to  furnish  even 
an  approximation  to  the  values  of  the  unknown  quantities. 

Adding  the  third  equation  to  the  first,  and  subtracting  the 
second  from  the  sum,  the  following  equation  results: 

0  =  4-  1.2"  -  0.83  (i)  4-  2.79  (3)  -  1.96  (4)  -  0.66  (5) 
4-  0.77  (6)  -  0.1 1  (8)  -  2.33  (17)  +  3.40  (18) 
—  1.07  (19).  (4) 

It  is  evident  that  if  the  first,  third,  and  fourth  equations  are 
satisfied,  the  second  must  also  be  satisfied.  Hence,  the  fourth 
equation  may  be  safely  substituted  for  the  second,  or,  as  will 
easily  appear,  for  the  first,  if  it  be  preferred  to  retain  the  second. 

The  fourth  equation  corresponds  to  the  symbol  (T..)-l^\  T.- 
T.  R.-Sp.,  and  might  have  been  obtained  directly  in  the  usual 
way.  The  second  suggestion  in  Art.  154.  to  use  small  angles 
once,  and  only  once,  would  have  led  to  the  selection  of  the 
fourth  equation  in  the  place  of  the  second. 


250  THE    ADJUSTMENT    OF    OBSERVATIONS 

The  second  suggestion  of  Art.  154  may  be  carried  out  still 
more  fully  by  so  selecting  one  of  the  side  equations  as  to 
involve  the  small  angles  of  the  triangle  Sp.-T.  R.-F.  T.  Let 
the  side  equation,  represented  by  the  symbol  (T.  R.)-Sp.-L.- 
F.  T.,  be  used.     It  is  : 

0=  -  4.0"  -I-  1.96(1)-  11.89(3)  +  9.93(4)  +  1.71(10) 

-2.91(11)  +  1.20(12)  +  9.27  (17)  -  11.60(18)  +  2.33  (19). (5) 

To  carry  out  the  third  suggestion  of  Art.  154,  that  it  is 
sometimes  desirable  to  use  a  side  equation  of  large  scope,  let 
(L.)-Sm.-F.  T.-Sp.-T.  R.  be  used.     It  is  : 

o  =  +  3.1"  -  0.83  (i)  +  2.79  (3)  -  1.96  (4)  -  0.66  (5)  +  2.72  (6) 
-  2.06  (7)  +  1.43  (13)  -  4.38  (14)  +  2.95  (16) 

+  3.40  (18)  -  4.33  (19)  +  0.93  (20).  (6) 

The  three  side  equations  recommended  as  best  are,  then,  the 
sixth,  third,  and  fifth. 

It  is  interesting  to  note  that  the  sixth  equation  is  the  same 
as  the  first  minus  the  second. 

181.  Length,  Azimuth,  Latitude  and  Longitude  Condition 
Equations.  —  In  the  adjustment  of  a  triangulation,  it  has  been 
shown  thus  far  how  in  a  net  joining  several  stations  the  condi- 
tions arising  from  the  closure  of  triangles  and  from  the  equality 
of  lengths  or  sides  as  computed  by  different  routes  can  be 
satisfied.  Cases  frequently  arise  in  which  other  condition  equa- 
tions must  be  satisfied,  in  addition  to  the  angle  and  side  equa- 
tions.    These  will  now  be  treated  briefly. 

The  principal  cases  which  arise  are  the  following  three,  and 
various  combinations  of  them  : 

I.  A  section  of  triangulation  which  starts  from  a  line  which 
is  fixed  in  length,  either  by  direct  measurement  or  by  previous 
triangulation,  may  end  on  a  line  which  is  similarly  fixed  in 
length.  In  this  case  a  length  condition  equation  must  be  used 
to  make  the  length  of  one  of  these   fixed  lines  as  computed 


ADJUSTMENT   OF  A  TRIANGULATION  — DIRECTIONS     251 

through  the  adjusted  triangulation  from  the   other   fixed   hue 
agree  with  its  fixed  value. 

2.  A  section  of  triangulation  may  include  two  lines  which 
are  each  fixed  in  azimuth,  either  by  previous  triangulation,  or  by 
astronomic  observations,  which  in  the  particular  case  furnish  a 
determination  of  the  azimuth  stronger  than  that  given  by  the 
triangulation.  It  is  then  necessary  to  use  an  azimuth  condition 
equation  to  insure  that  the  azimuth  as  carried  by  computation 
through  the  adjusted  triangulation  from  one  fixed  line  shall 
agree  at  the  other  fixed  line  with  the  azimuth  as  already  fixed 
there. 

3.  The  latitude  and  longitude  of  each  of  the  two  stations  in 
the  section  of  triangulation  may  have  been  fixed  by  previous 
triangulation,  and  it  may  be  desired  to  retain  these  positions 
unchanged.  This  may  be  done  by  writing  a  latitude  condition 
equation,  and  a  longitude  condition  equation,  such  as  to  insure 
that  the  latitude  and  longitude  as  computed  from  the  first  fixed 
station  through  the  adjusted  triangulation  shall  agree  at  the 
second  fixed  station  with  the  fixed  latitude  and  longitude  there. 

As  a  belt  of  triangulation  is  gradually  extended,  it  frequently 
happens  that  it  returns  upon  itself  in  such  a  manner  as  to  form 
a  complete  circuit.  In  such  a  case  it  usually  happens  that 
before  the  circuit  is  closed  by  the  field  operations  a  considerable 
portion  of  the  triangulation  in  the  circuit  has  been  adjusted,  the 
results  used  in  various  ways,  and  perhaps  published.  It  is  then 
desired  to  adjust  the  last  portion  of  the  circuit,  that  is,  the  por- 
tion still  unadjusted  when  the  last  of  the  field  work  is  done,  so 
that  no  discrepancies  of  any  kind  remain  at  the  closing  line.  In 
this  case  it  is  necessary  to  use,  in  this  last  portion,  in  addition 
to  the  angle  and  side  equations,  one  condition  equation  of  each 
of  the  other  kinds. 

In  any  case,  when  the  necessary  condition  equations  referring 
to  length,  azimuth,  latitude  and  longitude  have  been  formed, 
they  may  be  placed  with  the  angle  and  side  condition  equations, 
and  the  formation  of  correlates  and  of  normal  equations  and  the 


252  THE    ADJUSTMENT    OF    OBSERVx\TIONS 

solution  of  the  normal  equations  may  be  made  as  indicated  in 
Art.  179. 

182.  Length  Condition  Equations.  —  A  length  condition 
equation  is  written  in  the  same  form  as  a  side  equation,  and  is 
essentially  of  the  same  character.  It  serves  to  insure  that  the 
length  of  the  second  of  the  two  lines  which  are  fixed  in  length 
by  direct  measurement,  or  by  other  triangulation,  shall,  as  com- 
puted through  the  adjusted  triangulation  from  the  first  fixed 
line,  agree  with  its  fixed  value.  The  form  of  the  equation  is, 
when  expressed  in  terms  of  angles  : 

in  which,  in  a  selected  chain  of  triangles  from  the  first  to  the 
second  fixed  line,  A^,  A.,,  A^  .  .  .  is  in  each  case  the  angle 
opposite  the  required  side,  and  B^,  B.,,  B^  .  .  .  is  the  angle 
opposite  the  known  side.*  {A^),  {A,),  (A^)  .  .  . ,  (B^),  (B.^, 
(B.,)  .  .  . ,  are  the  required  corrections  to  these  angles.  B^^, 
8j,  S^  .  .  .,  hj..  hji^,  Sj^  .  .  .,  are  the  differences  for  i''  in 
the  logarithmic  sines  of  the  angles  A^,  A.,,  A^  .  .  . ,  B^,  B^^ 
B  .  .  .  .  (log  a)  is  the  necessary  correction  to  the  logarithm  of 
the  second  fixed  side,  a,  as  computed  through  the  selected  chain 
of  triangles  from  the  first  fixed  side,  b,  by  the  law  of  the  propor- 
tion of  sines,  using  the  uncorrected  angles  A^,  A,,  A^  .  .  . ,  i?,, 
B.^,  B^  .  .  . ,  to  make  it  agree  with  the  fixed  value  of  log  d. 

The  proof  that  the  formula  given  above  is  a  proper  expression 
of  the  length  condition,  is  similar  to  that  given  in  Art.  147, 
for  the  form  of  the  side  condition  equation  there  derived. 

To  express  this  equation  in  terms  of  corrections  to  directions, 
all  that  is  necessary  is  to  substitute  in  each  case  for  (A^),  (A.), 
(^3)  .  .  .,  (B^),  (B,),  (^3)  .  .  .,  which  are  corrections  to  angles, 
the  corresponding  differences  of  two  corrections  to  directions,  if 
neither  side  of  the  angle  is  fixed,  or  one  correction  to  a  direc- 

*  These  angles  A^,  A,,  A^  .  .  .,  B„  Bj,  B,,  .  .  . ,  may  appropriately  be 
called  distance  angles.  Their  locations  in  a  chain  of  triangles  are  illus- 
trated in  Fig.  25. 


ADJUSTMENT    OF    A   TRIAXGULATION —DIRECTIONS      253 

tion  with  the  proper  sign,  if  one  side  of  the  angle  is  a  Hne  fixed 
in  direction. 

Any  chain  of  triangles  between  the  fixed  lines  may  be  selected. 
It  is  well  to  select  a  chain  containing  a  minimum  number  of 
triangles,  thereby  making  the  number  of  terms  in  the  condi- 
tion equations,  as  small  as  possible,  and  reducing  the  labor  of 
solution.  It  is  well,  also,  in  order  to  avoid  instability  in  the 
solution  of  the  normal  equations,  to  select  the  strongest  possible 
short  chain,  that  is,  a  chain  in  which  small  angles  are  avoided  as 
far  as  possible. 

The  number  of  independent  length  condition  equations  in  any 
figure  is  one  less  than  the  number  of  lines  which  are  fixed  in 
length.  As,  for  example,  if  there  are  three  fixed  lengths  in  a 
figure,  and  length  condition  equations  are  written  fixing  the 
ratios  of  the  first  and  second,  and  first  and  third,  of  the  fixed 
lines  as  computed  through  the  adjusted  triangulation,  the  ratio 
of  the  computed  lengths  of  the  second  and  third  lines  is  thereby 
fixed.  Any  third  length  condition  equations  will  therefore  be 
derivable  from  the  condition  equations  already  written,  not 
independent  of  them. 

The  rigorous  adjustment  is  made  by  adding  the  length  con- 
dition equation,  or  equations,  to  the  other  condition  equations, 
proceeding  with  the  formation  of  correlates  and  normal  equations 
as  indicated  in  Art.  179.  Various  approximate  methods  of 
adjustment  have  been  proposed  and  used  in  the  place  of  the 
rigotpus  adjustment.  The  experience  of  the  Coast  and  Geodetic 
Survey  indicates  that  it  is  seldom  advisable  to  use  any  of  these 
approximate  solutions,  other  than  those  of  the  nature  indicated 
in  Art.  186. 

183.  Azimuth  Condition  Equations.  —  An  azimuth  condi- 
tion equation  serves  to  insure  that  the  azimuth  of  the  second  of 
the  two  lines  which  are  fixed  in  azimuth,  by  observations 
external  to  the  section  of  triangulation  being  adjusted,  shall,  as 
computed  through  the  adjusted  triangulation  from  the  first  fixed 
line,  agree  with  its  fixed  value. 


254  THE    ADJUSTMENT    OF    OBSERVATIONS 

Let  the  selected  chain  of  triangles,  for  a  section  of  triangula- 
tion  used  in  writing  the  length  condition  equation  as  indicated 
in  the  preceding  article,  be  illustrated  by  Fig.  26.  Let  it 
be  supposed  that  the  order  of  computation  is  that  indicated 
by  the  numbering  of  the  triangles  i  to  6.  The  distance  angles 
used  in  forming  the  length  condition  equation  are  marked  by 
the  letters  A  and  B.  Let  it  be  supposed  that  the  lines  DE  and 
NO  are  fixed  in  azimuth. 

For  the  same  reason  that  in  forming  the  length  condition 
equations  the  number  of  terms  was  kept  as  small  as  possible, 
namely,  to  save  work  in  the  computation,  so  here  the  azimuth 
condition  equation  should  directly  involve  as  few  angles  as 
possible.  It  is  possible  to  carry  the  azimuth  by  computation 
through  a  chain  of  triangles  by  using  only  one  angle  in  each 
triangle  (the  lengths  being  supposed  known),  namely,  the  angle 
marked  C  in  Fig.  26.  This  third  angle  in  each  triangle  will, 
for  convenience,  be  called  the  azimuth  angle  to  distinguish  it 
from  the  distance  angles.  The  azimuth  condition  equation 
should  involve  the  azimuth  angles  only  in  the  selected  chain  of 
triangles  through  the  figure  used  in  writing  the  length  equation. 

As  a  given  correction  to  any  of  the  azimuth  angles  in 
Fig.  26  affects  the  azimuth  of  the  line  NO  as  computed 
through  the  adjusted  triangulation  by  precisely  the  amount  of 
correction  to  that  angle,  it  is  evident  that  the  general  form  of 
the  azimuth  condition  equation  is 

0=-(a)+2(Q)-:S(Ci), 

in  which  (a)  is  the  correction  to  the  azimuth  of  the  second  fixed 
line,  as  computed  through  the  unadjusted  azimuth  angles  from 
the  first  fixed  line,  necessary  to  make  it  agree  with  the  fixed 
azimuth  there,  and  2  Cj^)  is  the  sum  of  the  corrections  to  the 
azimuth  angles  which  are  on  the  right-hand  side  of  the  chain  of 
triangles  when  proceeding  in  the  direction  of  computation, 
and  S  (Cz)  is  the  sum  of  corrections  to  the  left-hand  azimuth 
angles. 


ADJUSTMENT   OF  A  TRI ANGULATION  — DIRECTIONS     255 

For  example,  in  Fig.   26  the  azimuth  condition  equation  is 
o  =  -  (a)  +  (CJ  -  (Q  -  (Q  +  (Q)  +  (Q  +  (Q, 
in  which  (a)  is  the  fixed  azimuth  of  NO  minus  its  azimuth  as 
computed  through   the    unadjusted   angles  C,,   C^   Cj,   .  .  .   Cg 
from  the  fixed  azimuth  of  DE. 

To  express  this  condition  equation  in  terms  of  directions,  it  is 
necessary  simply  to  substitute  for  each  correction  to  an  angle 
the  difference  of  the  two  corrections  to  directions,  one  of  which 
is  necessarily  zero  if  that  particular  direction  is  already  fixed. 

As  the  number  of  length  condition  equations  is  one  less  than 
the  number  of  fixed  lengths  in  a  figure,  so  the  number  of  azimuth 
condition  equations  is  one  less  than  the  number  of  fixed 
azimuths. 

184.  Latitude  and  Longitude  Condition  Equations. — A 
latitude  condition  equation  serves  to  insure  that  the  latitude  of 
the  second  of  the  two  points 
which  are  fixed  in  latitude,  by 
observations  external  to  the 
triangulation  being  adjusted, 
shall,  as  computed  through  the 
adjusted  triangulation  from  the 
first  of  two  points  fixed  in  lati- 
tude, agree  with  its  fixed  value. 
A  longitude  condition  equa- 
tion serves  the  same  purpose 
with  respect  to  longitudes. 

In  Fig.  26  the  latitude  and 
longitude  of  each  of  the  points  E  and  N  are  supposed  to  be 
fixed.  It  is  supposed  that  the  position  <^„X„  of  point  N  as 
computed  through  the  unadjusted  triangulation  differs  from  its 
fixed  position  </>„A„,.  It  is  required  to  write  the  latitude  and 
longitude  condition  equations  necessary  to  remove  the  dis- 
crepancy. 

In  the  figure  the  azimuth  angles  have  been  marked  C, 'ind  the 
distance  angles,  of    two  classes,  are  marked  A  and  B,  the  A 


256  THE    ADJUSTMENT    OF    OBSERVATIONS 

angle  being  in  each  case  opposite  the  required  side,  and  the 
B  angle  being  opposite  the  known  side.  The  computation  is 
supposed  to  proceed  from  DE  to  ON. 

The  azimuth  is  supposed  to  have  been  computed  through  the 
angles  C. 

Let  S„  hfi  be  the  logarithmic  sine  differences  for  \"  for  an 
A  distance  angle  and  a  B  distance  angle. 

Let  71/ be  the  modulus  of  the  common  system  of  logarithms. 

B  A 

Let  ^\=  —r  ^°^  ^''  ^^^  ^"  ^"^  ''2  ^  ~d  ^^^  ^"  ^^^  ^"' 

■^  It  -^  c 

In  the  expressions  for  7\  and  ;;,  A„  is  the  value  at  the  point 

N  of 

(1—^2  sin^  </))^ 

A  = 77 > 

a  arc  i 

and  B^  is  the  value  at  the  vertex  of  any  C  angle  of 

(i  -  ^"  sin^  <ti)i 

-D   =  ; ^^ 77    • 

a  {1  —  e-)  arc  i 
In  these  expressions  for  A  and  B,  a  is  the  equilateral  radius  of 
the  spheroid  on  which  the  triangulation  is  computed,  and  e  is 
its  eccentricity.  A  and  B  are  factors  used  in  the  computation 
geodetic  positions.  For  their  values  on  the  Clark  Spheroid  of 
1 866,  see  Appendix  9  of  the  Coast  and  Geodetic  Survey  Report 
for   1894. 

In  each  triangle,  reckoning  all  the  angles  in  seconds,  a  cor- 
rection {A)  in  a  distance  angle  A  will  produce  a  correction  in 

the  computed  latitude  at  N  of  '  ~  M  '  ^"^  ^  correc- 
tion in  a  computed  longitude  of  — ^jrj^ •  A  correc- 
tion to  a  distance  angle  B  produces  similar  corrections  to  the 
latitude  and  longitude  at  N,  but  with  the  reverse  algebraic  sign. 

<^,.X,  is  the  position  of  the  vertex  of  the  C  angle  in  the 
triangle  containing  the  angle  which  is  supposed  to  be  corrected. 

Let  any  azimuth  angle  on  the  right  side  of  the  chain  of 
triangles  proceeding  in  the  direction  of  progress  of  computation 


ADJUSTMENT    OF   A    TRIANGULATION  —  DIRECTIONS     257 

be  designated  by  Cp^,  and  any  such  angle  on  the  left  side  by  C/^. 
Then  any  correction  to  a  right  azimuth  angle  produces  a  correc- 
tion in  latitude  at  N  of  +  i\  (X„  —  \)  {Crj  and  in  longitude 
of  —  r,  (<^„  —  ^<.)  (C/.).  Similarly,  any  correction  to  a  left 
azimuth  angle  produces  the  corrections  —  i\  (\„  —  \.)  (C/,)  and 
+  ^'2  (0«  —  <^c)  (C/,)  to  the  latitude  and  longitude  respectively 
at  A''.*     Hence, 

+  r,  (A„  -  A,)  iCj^  -  /-  (A„  -  A,)  (cA  (i) 

„  _  .         .      ,    ^  r(A„-A.)g^(^)       (A„  -  A.)  In  jB) 
o  -  A„  -  A„-  +  2  [^ j^ -^ 

-  r,  (</,„  -  c^,)  {Cn)  +  r,  (<A„  -  <Ae)  {CiA  (2) 

With  sufficient  accuracy  and  greatest  convenience  </>„  —  <\>^ 
and  X„  —  X^  may  usually  be  reckoned  in  minutes  and  tenths. 

Then,  multiplying  the  equations  by  —  =  (I2::^ibi)  in  order  to  re- 

60       \    60   / 

move  a  constant  factor  from  the  coefficients  of  [A)  and  {B), 

o  =  .00724  (,^„  -  <^,/)  +  2  [(<^„  -  <^,)  Ki  {A)  -  (<^„  -  c^,)  8yj  {B) 

+  -4343  ^1  iK  -  K)  (Cn)  -  .4343  (K  -  K)  (Cl)1  (3) 

o  =  .00724  (A„  -  A„0  +  2  [(A„  -  A,)  8a  (A)  -  (A„  -  A,)  Sy,  (B) 

-  -4343  ^'2  (^n  -  *^c)  (Of)  +  .4343  ^2  (</»«  -  *^.)  (<^/-)]-        (4) 

185.  If  these  condition  equations  are  used  in  this  form,  and 
if  there  is  a  length  condition  equation  and  an  azinnith  condition 
equation  covering  the  same  chain  of  triangulation,  it  will  be 
found  tiiat  the  normal  equations  have  large  side  coefficients,  and 
their  solution  will  be  unstable.  The  following  transformation 
facilitates  the  solution  by  avoiding  this  difficulty. 

*  To  sliow  the  derivation  of  llie  formula;  for  /•,  and  /:,  would  rcciuire 
considerable  space.  This  derivation  has  no  bearing  upon  the  method  of 
least  squares.  It  suffices  for  the  present  purpose  to  note  that  ;■,  and  r^  are 
functions  of  the  latitude,  and  that  the  formula'  ji^iven  show  tlic  relation  be- 
tween (C)  and  the  corresponding  corrections  in  latitude  and  l<)iiL;iliidc  at  yV. 


258  THE    ADJUSTMENT    OF    OBSERVATIONS 

Assume  a  point  and  (/>/>;,  approximately  in  the  mean  latitude 
and  longitude  of  the  points  <f),\  which  are  at  the  vertices  of 
the  C  angles.  Multiply  the  length  condition  equation  given  in 
Art.  152,  which  is  in  the  form 

o  =  (log  a)  +  2  Sa  (A)  2  8n  (B),  by  c/>,  -  c^,„ 

and  the  azimuth  condition  equation  given  in  Art.  153,  namely, 

o  =  -  (u)  +  2  (Cji)  -  2  (C^),  by  .4343  '\  (K  -  Ky 

and  add  the  products  to  equation  (3).  Also  multiply  the  length 
equation  by  (X/,  —  \,)  and  the  azimuth  equation  by  —  .4343  r^ 
(<^^  _  <^J  and  add  the  products  to  equation  (4).  Then  the 
latitude  and  longitude  condition  equations  are  as  follows,  in 
which  all  that  portion  within  the  first  square  bracket  in  each 
case  constitutes  the  absolute  term : 

o  =  [.00724  (<^„  -  <^„')  (c^A  -  <^«)  (log  a)  +  .4343  n  (K  -  K)  (a)] 

+  2  [(<t>n  -  <^o)  ^A  (A)  -  (<f>,  -  <^,)  Bn  (B) 

+  .4343  r,  (A;,  -  K)  (Cn)  -  .4343  (K  -  K)  (Cl)].  (s) 

o  =  [.00724  (A,.  -  A,/)  (A^  -  AJ  (log  .7)  -  .4343  f\  (cj>,,,  -  <^„)  (a)] 

+  2  [(A,  -  A,)  8.,  (A)  -  (A;,  -  A,)  Ss  (B) 

-  .4343  ^2  (*k  -  ^c)  (Cii)  +  .4343  r,  (<!>,  -  4>c)  (Cl)].  (6) 

This  transformation  is  allowable  since  the  adjusted  angles 
which  satisfy  the  length  and  azimuth  equations  and  equations 
(5)  and  (6)  must  evidently  satisfy  (3)  and  (4)  also,  and  no  new 
condition  has  been  put  into  the  solution,  as  (5)  and  (6)  are  mere 
combinations  of  the  original  conditions. 

The  transformation  has  the  effect  of  transferring  the  point 
at  which  the  discrepancy  in  latitude  and  longitude  is  supposed 
to  have  developed  from  the  end  of  the  triangulation  ^„X„  to  a 
mean  point  4>h\-  Equations  (3)  and  (4)  correspond  to  the 
supposition  that  the  computation  of  latitudes  and  longitudes  has 
progressed  continuously  from  the  beginning  of  the  section  of 
triangulation  being  adjusted  to  (^„X„.  Equations  (5)  and  (6) 
correspond  to  the  supposition  that  the  computation  of  latitudes 
and  longitudes  has  been  made   in   two  sections,  one  from  the 


ADJUSTMENT    OP    A    TRIANGULATION  —  DIRECTIONS     259 

beginning  of  the'  section  being  adjusted  to  (f),,\/,:  and  the  other 
from  the  end  of  the  section  (at  (}>n\)  back  to  (f>h\,  this  portion 
of  the  position  computation  being  supposed  to  be  made  by 
starting  with  the  fixed  length  and  azimuth  found  at  a  Hne  of 
which  (f>„\  is  one  end ;  and  that  the  discrepancy  in  latitude  and 
longitude  is  developed  at  (f>/,\  by  the  junction  of  the  two 
position  computations  there. 

To  express  the  latitude  and  longitude  condition  equations  as 
written  above  in  terms  of  corrections  to  directions,  it  is  necessary 
simply  to  substitute  for  each  correction  to  an  angle  the  differ- 
ence between  two  corrections  to  directions,  one  of  which  is 
necessarily  zero  if  that  particular  direction  is  fixed. 

The  number  of  latitude  condition  equations  in  a  figure  is  one 
less  than  the  number  of  groups  of  points  fixed  in  latitude,  each 
group  being  composed  of  points  which  are  tied  together  by 
lines  already  fixed  in  length  and  azimuth,  and  being  separated 
from  other  groups  by  lines  not  so  fixed.  The  removal  of  the 
latitude  discrepancy  for  one  point  of  such  a  group  removes  it 
for  all.  Similar  statements  are  true  for  the  longitude  condition 
equations. 

Ou  the  Breaking  of  a  Net  of  Triangulation  into  Seetions  for 
Convenience  of  Solution. 

186.  In  a  long  chain  of  triangulation,  or  in  a  complicated  net, 
the  simultaneous  solution  of  the  condition  equations,  which  are 
required  by  theory  to  secure  the  ideal  most  i:)robable  results, 
would  be  very  troublesome,  not  from  any  principle  involved,  but 
from  its  very  unwieldiness.  For  example,  such  a  simultaneous 
solution  for  all  the  primary  triangulation  now  forming  a  con- 
tinuous net  in  the  United  States  would  involve  more  than  two 
thousand  condition  equations.  Accordingly  it  is  necessary  to 
break  the  triangulation  into  sections  and  adjust  each  section  by 
itself.  As  this  breaking  into  sections  causes  more  or  less  dis- 
turbance of  the  ideal  solution  at  or  near  the  lines  of  breaking, 
the  exercise  of  judgment  is  required  in  the   selection  of  these 


26o  THE    ADJUSTMENT    OF    OBSERVATIONS 

lines.  The  larger  the  sections  are  made,  the  nearer  the  approxi- 
mation to  the  ideal  solution,  and  on  the  other  hand  the  greater 
will  be  the  labor  of  computation. 

The  present  practice  in  the  Coast  and  Geodetic  Survey  is  to 
divide  the  triangulation  into  sections,  adjust  each  section  by  the 
rigorous  method  set  forth  in  this  chapter,  and,  in  commencing 
the  adjustment  of  each  new  section,  to  hold  as  absolutely  fixed 
in  all  respects  all  lines  which  have  entered  into  a  previous 
adjustment.  This  method  has  the  great  advantage  of  giving  the 
final  results  by  a  single  adjustment.  It  is  believed  that  with  a 
proper  selection  of  lines  of  separation  betwoen  adjustments  the 
results  obtained  are  so  close  an  approximation  to  the  ideal  best 
results  that  no  further  expenditure  of  energy  in  computation  is 
warranted.  Two  principles  guide  in  selecting  the  lines  of 
separation.  First,  such  a  line  should  be  one  which  is  strongly 
determined.  Second,  it  should  be  a  line  which  enters  in  but 
few  conditions  in  one  of  the  two  sections.  For  example,  a  line 
at  the  edge  of  a  base  net  and  forming  the  first  line  of  the  chain 
of  triangulation  connecting  that  base  net  with  the  next  is 
frequently  selected  as  a  line  of  separation.  It  is  strongly  deter- 
mined in  length  by  the  base  and  base  net.  It  is  usually 
involved  in  but  few  conditions  on  the  side  toward  the  chain  of 
triangulation.  In  the  separation  into  sections  in  primary  tri- 
angulation the  most  frequent  cases  are: 

1.  A  section  which  begins  with  a  line  at  the  margin  of  one 
base  net  and  extends  to  a  line  at  the  far  side  of  the  next  base 
net,  thus  including  the  second  base  net. 

2.  A  section  which  includes  two  base  nets  and  the  triangula- 
tion between  them. 

3.  A  section  which  is  limited  at  each  end  by  a  line  fixed  by 
previous  adjustment  in  azimuth,  latitude,  and  longitude  as  well 
as  in  length. 

In  secondary  and  tertiary  triangulation  a  great  variety  of 
cases  arise,  and,  in  general,  the  adjustment  is  made  in  smaller 
sections  than  the  primary  triangulation. 


CHAPTER    VIII 

APPLICATION    TO    BASE-LINE    MEASUREMENT    AND    TO    LEVELING 

187.  Precision  of  a  Base-Line  Measurement.  —  For  clear- 
ness it  will  be  necessary  to  outline  the  principles  on  which  the 
measurement  is  made. 

First,  we  must  find  the  length  of  the  measuring  bar  or  tape 
in  terms  of  some  standard  of  length;  and  as  the  measurements 
of  the  line  itself  are  made  at  various  temperatures,  the  coeffi- 
cients of  expansion  of  the  metals  in  the  measuring  apparatus 
must  also  be  known.  Comparisons  must,  therefore,  be  made 
with  the  standard  during  wide  ranges  of  temperature;  and  as 
these  comparisons  are  fallible,  the  results  found  for  length  and 
expansion  will  be  more  or  less  erroneous. 

The  principle  involved  in  the  measurement  is  exactly  the 
same  as  in  common  chaining  with  chain  and  pins.  There  are, 
indeed,  various  contrivances  for  getting  a  precision  not  looked 
for  in  chaining,  such  as  for  aligning  the  measuring  bar,  for 
finding  the  inclination  of  each  position  of  the  bar,  and  for  estab- 
lishing fixed  points  for  stopping  at  and  starting  from  in  measure- 
ment.    But  these  make  no  change  in  the  essential  principle. 

The  errors  in  the  value  of  a  base  line  may,  therefore,  be  con- 
sidered to  arise  from  two  principal  sources,  —  comparisons  and 
measurement.  Experience  has  shown  that  a  considerable 
portion  of  the  error  in  the  length  of  a  base  arises  from  the  errors 
in  the  comparisons  which  serve  to  determine  the  length  of  the 
measuring  bar  or  tape. 

These  errors  differ  essentially  in  character.  An  error  arising 
from  the  comparisons,  being  the  same  for  each  bai-  measure- 
ment, is  cumulative  for  the  whole  base,  while  errors  arising  in 
the   measurement   of    the   base   itself,  were   the   measurements 

261 


262  THE    ADJUSTMENT    OF    OBSERVATIONS 

repeated  often  enough  and  the  conditions  sufficiently  varied, 
would  tend  to  mutually  balance,  and  could,  therefore,  be  treated 
by  the  strict  principles  of  least  squares.  But  as  the  number  of 
measurements  is  not  often  more  than  2  or  3,  and  as  these  are 
made  usually  at  about  the  same  season  of  the  year,  only  a  com- 
paratively rough  estimate  of  the  precision  is  to  be  looked  for. 

As  a  check  on  the  field  work  a  base  is  usually  divided  into 
sections  by  setting  stones  firmly  in  the  ground  at  approximately 
equal  intervals  along  the  line,  so  that  instead  of  being  able  to 
compare  results  at  the  end  points  only,  we  may  compare  results 
just  as  well  at  6  or  8  points.  In  this  way  a  better  idea  of  the 
precision  of  the  work  is  obtained,  as  we  have  6  or  8  short  bases 
to  deal  with  instead  of  a  single  long  one. 

We  proceed  now  with  the  problem  of  determining  the  pre- 
cision of  measurement.  It  may  be  stated  as  follows:  A  base  is 
measured  in  n  sections  with  a  bar  of  a  certain  length,  each 
section  being  measured  11^  times.  By  the  first  measurement  the 
first  section  contains  J//  bars,  the  second  M^'  bars,  .  .  .  ;  by 
the  second  measurement  the  first  section  contains  M,'  bars,  the 
second  J//'  bars,  .  .  .  ;  and  so  on.  The  weights  of  the  meas- 
urements in  opder  being  //,  //,  .  .  .  ;  p(' ,  p.'',  ...;... 
respectively,  required  the  m.  s.  e.  of  the  most  probable  value  of 
the  base. 

Let  x^  =  most  prob.  value  of  first  section, 

x^  =  most  prob.  value  of  second  section, 


then  we  have  the  observation  equations: 
First  section,  x^  —  J//   =  i\       wt.  //, 

x^  -  m;'  =  vi'   wt.  p;', 


Second  section,  x^  —  M^'   =  r^'      wt.  //, 

x^  —  M^'  =  v,y     wt.  //', 


and  so  on. 

Now,  either  of  two  assumptions  may  be  made. 


APPLICATION    TO    BASE-LIXE   MEASUREMENT  263 

188.  {a)  In  the  first  place,  that  the  precision  of  the  measure- 
ment of  each  bar  length  is  the  same  throughout  the  different 
sections. 

We  have,  then,  iin^  equations  containing  n  unknowns,  and  the 
normal  equations  are 

+  [A]-^-i  =[P,^'Q, 


whence  x^^  x.„   .  .  .   are   known,   and  therefore  the  whole  line 
X  =  x^  -{-  x.^  -\-  .   .   .  -\-  .f„  is  known. 

The  probable  error  r  of  an  observation  of  weight  unity  — 
that  is,  of  a  single  measurement  of  a  bar  length  —  is  given  by 
(see  Art.  105) 


.=  0.6„5V/  ^'-'^ 


No.  of  obs.  —  No.  of  indep.  unknowns 


=  o.6„5v/:    [^^J 


91  {?l,   -    l) 


Now,  the  length  of  the  measuring  bar  being  taken  as  the  unit 
of  measurement,  the  weight  of  a  section,  as  depending  on  the 
measurement,  may  be  expressed  in  terms  of  the  number  of  bar 
lengths  measured.  For  since  r  is  the  p.  e.  of  a  measurement 
of  a  single  bar  length,  the  p.  e.  of  the  measurement  of  a  length  of 

J/ bars  is  rVJ/.     Hence  -—  is  the  weight  of  a  measurement 

of  length  M  when  the  weight  of  a  measurement  of  the  unit  of 
length  is  unity. 

Writing,  therefore,  for  the  weights  /  their  values  in  terms  of  Jl/, 


=  °-^^^^\n(n,'-.)\%\' 


In  the  case  usually  occurring  in  practice,  where  the  line  is 
measured  twice,  we  may  put  this  formula  in  a  form  more  con- 
venient for  computation.     For  if  the  first  measurement  of  the 


!^4 


THE    ADJUSTMENT    OF    OBSERVATIONS 


«j  sections  gave  lengths  M^,  M,^  ■  .  .  ,  and  the  second  meas- 
urements gave  lengths  31^  +  ^i,  ^^^  +  ^i^i  ...  for  the  same 
sections  in  order,  then,  since 


b'] 


'h[~. 


/y 


2 


we  have  for  the  m.  s.  e.  of  one  measurement  of  a  bar  length  and 
for  the  mean  of  two  measurements  respectively, 


v/^KlandV-rSl- 
V  2  ;/  \_Mj         2\  n  \_M\ 


Hence  the  p.  e,  of  a  single  measurement  and  of  the  mean  of  the 
two  values  of  the  whole  base  are 


0.67 


V    2  n 


Tf 


and  0.34  \/ 


'[J/] 


\P' 
M 


the  number  of  bar  lengths  in  the  line  being  [^1/]. 

Ex. — The  Bonn  Base,  measured  in  1847,  near  Bonn,  Germany,  with  the 
original  Bessel  metallic-thermometer  apparatus.  The  base  was  a  broken 
one,  the  two  parts  making  an  angle  of  179°  23'.  Each  part  was  measured 
twice  as  follows  :  * 


Differences. 

No.  OF  Bar 
Lengths. 

Northern  Part 

Sec.  I 

Sec.  2 

Sec.  3 

Southern  Part 

Sec.  I 

Sec.  2 

Sec.  3 

L 
-0.183 
+  0.094 

-  0.013 

—  0.007 
+  0.095 
+  0.757 

116 

87 
61 

264 

92 

60 

131 

—    2S3 

Hence  the  m.  s.  e.  of  the  northern  part,  arising  from  errors  of  measurement 
only,  is 


L 

0.093, 


I       /264  l.\%f       .0942        .oi32\  _ 
*  Das  rhei>tische  Dreiecks}ictz.     Berlin,  1876. 


APPLICATIOX    TO    BASE-LINE   MEASUREMENT         265 

and  the  m.  s.  e.  of  the  southern  part  is 

X       /28,roof^.o^^.-jyr\ 
2    V     3     \   92  60  131  / 


L 

0.327. 


The  other  two  main  sources  of  error  are: 

1.  Error  in  comparison  of  the  measuring  bars  with  one  another. 

2.  Error  in  the  determination  of  their  length. 

The  m.  s.  e.  arising  from  these  sources  are,  respectively, 

L  L 

rt  0-386,      J^  0-313  for  the  northern  part, 
i  0.391,      ^  0.335  fo""  ^^^  southern  part. 

Remembering  that  these  latter  errors  are  systematic,  we  have,  finally, 
p.  e.  of  base  =  .6745  ^.093^  +  -327^  +(-386  +  .391)'  +  i-Z'^l  +  -335)''' 

L 

=  0.72. 

189.  {b)  In  the  second  place,  if  \vc  assume  that  the  law  of 
precision  of  the  measurements  of  the  different  sections  is 
unknown,  and  that  these  sections  are  independent,  we  have  for 
the  mean  of  the  values  of  the  several  sections  and  their  m.  s.  e., 


X, 


[A] 


^J  =  A^-.  =       y^^     ,  since//  =  A"  =  •  •  •  =  ^ 


If  X  denotes  the  whole  line,  so  that 

X  =  x^  +  X.,  +    •  -  -    -f  .T„, 
then,  since  the  measurements  are  indci)endcnt. 


266  THE    ADJUSTMENT    OP    OBSERVATIONS 

and  the  (m.  s.  e.)'"  of  a  single  measurement  of  the  Une 

The  number  of  bar  lengths  m  the  line  being  [v^/],  we  have  for 

the  average  value  of  the  (m.  s.  e.)^  of  a  single  measurement  of  a 

bar  length 

I       [r^]  +  [V,']  +   •  •  • 
«j  -  I  [M] 

If,  for  example,  the  line  has  been  measured  twice,  and 
d^,  d,,  .  .  .  d„  are  the  differences  of  the  measurements  of  the 
several  sections,  then 

—  I 

2 


[^v 


7,  21    = 


d' 


and  therefore 


l^v  = 


_[^2] 


and   the   (m.  s.  e.)^  of  a   single   measurement   of  a  bar  length 
I    [./^'] 


IS 


2     [J/] 


Ex.  —  The  Chicago  Base,  measured  in  1877  with  the  Repsold  metaHic- 
thermometer  apparatus  belonging  to  the  United  States  Engineers.  The 
base  was  divided  into  8  sections,  and  was  measured  twice. 


Section. 

No.  OF  Bar  Lengths. 

DiF.  OF  Measures. 

vim. 

I. 

227.25 

-  1-3 

II. 

230.25 

+  2.5 

III. 

234-50 

+  2.3 

IV. 

232 

+  0.7 

V. 

231 

+  1-5 

VI. 

225 

+  i-i 

VII. 

300.50 

+  1-3 

VIII. 

iq6.8o 

—  0.2 

APPLICATION    TO    BASE-LINE   MEASUREMENT         267 

Taking  the  errors  of  the  different  sections  as  independent,  the  p.  e.  of  the 
mean  of  the  two  measures  of  the  base  is 

0-6745  y-j^=  I-46. 

The  p.  e.  arising  from  the  other  sources  of  error  were 

}iitii. 
(i)  Measuring  bar it  6.38, 

(2)  Metallic  thermometer  .      .      .      .     :^  2  .?>2, 

(3)  Elevation  above  mean  tide,  N.Y.     rt  0-36. 

Assuming  these  to  be  independent,  the  p.  e.  of  the  Chicago  Base  at  sea 
level  is 


V1.462  +  6.38-  +  2.82-  +  0.36'  =  7.14. 
APPLICATION    TO    LEVELING. 

190.  Lines  of  levels  are  usually  run  in  duplicate,  each  por- 
tion beincr  leveled  over  twice,  sometimes  both  leveling:s  beine:  in 
the  same  direction,  but  preferably  in  opposite  directions.  The 
probable  error  of  the  mean  result  for  a  single  kilometer,  or  for 
the  whole  line,  may  then  be  computed  from  the  discrepancy 
between  the  two  levelings  over  each  section  by  the  formulas 
given  in  the  preceding  article,  in  connection  with  base  line 
measurements. 

Whenever  a  series  of  such  lines  becomes  so  connected  as  to 
form  a  network,  an  additional  determination  of  the  accuracy  of 
the  work  is  afforded  by  the  closing  errors  of  the  various  circuits 
forming  the  net.  Experience  shows  that  the  probable  errors  as 
thus  computed  from  the  adjustment  are  usually  considerably 
larger  than  as  computed  from  the  discrepancy  on  sliort  sections 
between  the  two  runnings  of  each  line,  thus  indicating  that  there 
are  systematic  errors  in  the  leveling  which  are  n(jt  eliminated  by 
duplicating  each  line. 

The  rigorous  adjustment  of  a  level  net  may  be  made  in  eitlu-r 
of  two  ways,  namely,  by  the  method  set  forth  in  Chapter  I\'  in 
dealing  with  indirect  observations,  or  by  the  method  of  Chapter 
V  which  is  applicable  to  conditioned  observation. 


268  THE    ADJUSTMENT    OF    OBSERVATIONS 

191.  Method  of  Indirect  Observations.  —  Let  it  be  supposed 
that  s  is  tlie  total  number  of  junction  bench-marks  in  a  net  of 
leveling,  each  of  these  junction  bench-marks  being  common  to 
three  lines  of  the  net.  Let  /  be  the  number  of  lines  in  the  net, 
each  connecting  two  junction  bench-marks.  Now  if  it  be 
assumed  that  the  unknowns  desired  are  the  elevations  of  each  of 
J  —  I  of  these  junction  bench-marks,  referred  to  the  remaining 
bench-mark  as  a  zero  point,  the  case  kn  hand  is  one  of  indirect 
observations  with  the  no  conditions.  There  are  s  —  i  unknowns, 
and  /  observation  equations  each  of  the  form 

X  —  J  =  /    or     X  =  /, 

see  equations  (2)  of  Art.  yy,  and  the  solution  is  carried  otit  in 
accordance  with  Chapter  IV. 

192.  Method  of  Conditioned  Observations.  —  As  before,  let 
it  be  supposed  that  s  is  the  total  number  of  junction  bench-marks 
in  the  net  /  and  the  number  of  lines  each  connecting  two  junc- 
tion bench-marks.  If  it  be  assumed  that  the  required  unknowns 
are  the  /  differences  of  elevations  between  the  junction  bench- 
marks at  the  ends  of  each  of  the  /  lines,  the  case  in  hand  is  one 
of  conditioned  observations.  There  will  be  /  observation  equa- 
tions each  of  the  form  .r  =  I  expressing  the  direct  observation  in 
each  case  of  one  of  the  required  unknowns.  The  number  of 
differences  necessary  to  fix  the  relative  elevations  of  s  bench- 
marks is  J-  —  I.  The  number  of  observed  differences  in  excess 
of  this  required  number  is  /  —  (j-  —  i),  and  therefore  the  num- 
ber of  condition  equations  is  /  —  ^  +  i.  These  conditions  exist 
as  the  requirement  that  each  circuit  in  the  net  must  close,  that 
is,  the  sum  of  the  differences  of  elevation  in  order  around  each 
circuit  must  be  zero.  The  solution  may  be  carried  out  as  indi- 
cated in  Chapter  V  by  the  Method  of  Correlates  (see  Art.  119). 

193.  One  question  which  arises  at  the  outset  with  either  of 
these  two  general  methods  of  solution  is,  what  relative  weights 
shall  be  assigned  to  the  different  lines.  If  all  the  leveling  is 
done  with  one    type  of   instrument  by  a  uniform  method  and 


APPLICATION    TO    LEVELING  269 

under  similar  conditions,  the  problem  of  assigning  weights  is 
simply  that  of  determining  the  relation  between  the  errors  in  the 
lines  and  their  lengths.  If  all  of  the  errors  are  of  the  acciden- 
tal class,  the  accumulated  errors  in  different  lines  tend  to  be  as 
the  square  roots  of  their  lengths,  and  therefore  the  proper 
weights  are  inversely  as  the  lengths.  If,  on  the  other  hand,  the 
principal  errors  are  of  the  systematic  class,  the  accumulated 
errors  are  proportional  to  the  length,  and  the  weights  to  be 
assigned  are  inversely  as  the  squares  of  the  length.  The  de- 
cision as  to  the  assignment  of  weights  must  be  based  on  the 
computer's  judgment  as  to  whether  the  lines  should  be  assigned 
to  one  or  the  other  of  these  classes,  his  judgment  being  based 
upon  special  investigation,  if  possible. 

194.  If  the  leveling  combined  in  a  net  is  done  with  various 
instruments  and  methods  and  under  strongly  contrasting  condi- 
tions, the  difficulty  of  assigning  proper  weights  is  largely  in- 
creased. In  such  a  case  the  basis  for  judgment  as  to  the  relative 
weights  which  should  be  assigned  will  be  frequently  found  to  be 
rather  insecure.  If  the  net  of  levels  is  extensive,  and  if  there  are 
several  lines  in  each  of  the  classes  of  leveling,  the  following 
principle  may  be  used  to  determine  whether  the  weights  assigned 
are  approximately  correct  and  what  modifications,  if  any,  are  de- 
sirable. As  the  correct  weights  are  inversely  proportional  to 
the  squares  of  the  probable  errors,  it  follows  that  in  general,  un- 
less there  is  an  extraordinary  difference  as  to  the  manner  in 
which  the  different  classes  of  lines  are  involved  in  the  net,  that 
the  average  value  of  pv'  for  each  class  of  lines  should  be  the 
same,  if  the  assigned  weights  are  correct.  If  a  certain  class  of 
lines  be  assigned  too  great  weight  in  a  gi\-en  adjustment,  the 
average /'■zr  for  that  class  will  be  larger  than  for  other  classes, 
and  vice  versa.  New  relative  weights  may  be  assigned  as  indi- 
cated by  the  average  values  of  pir  for  the  different  classes,  and 
a  new  adjustment  made  which  will,  in  turn,  furnish  a  new  test 
of  the  assigned  weights.  In  making  the  changes  in  weights,  it 
should  be  kept  in  mind  that  as  the  vveiglits, />,  for  a  given  class 


270  THE    ADJUSTMENT    OF    OBSERVATIONS 

are  decreased,  there  will  be  a  tendency  for  the  iPs,  in  that  class 
to  increase.  It  is  not  important  to  make  a  very  close  approxi- 
mation to  the  ideal  weights,  for  the  reason  that  it  will  be  found 
in  general  that  a  considerable  change  in  weights  is  accompanied 
by  only  a  small  change  in  the  numerical  results  of  the  adjust- 
ment. For  an  example  of  the  application  of  this  method  of 
testing  assigned  weights  in  connection  with  the  adjustment  of 
the  precise  level  net  of  the  United  States,  see  Appendix  8  of  the 
Coast  and  Geodetic  Survey  Report  for  1 899,  pp.  437, 438,447,448. 

The  same  principle  may  be  applied  to  testing  the  assigned  re- 
lation between  the  lengths  of  the  lines  and  their  weights.  If  the 
values  oi  p'-f  resulting  from  an  adjustment  for  the  various  lines 
of  a  single  class  of  leveling  are  tabulated  in  order  of  the  lengths 
of  the  lines,  the  values  should  show  no  progressive  change  if  the 
assigned  relation  between  length  and  weight  is  correct.  If,  for 
example,  on  the  other  hand,  weights  inversely  proportional  to 
the  length  have  been  assigned  to  a  given  class  of  leveling,  where- 
as they  should  have  been  inversely  proportional  to  the  square  of 
the  length,  it  will  be  found  that  the  average  pir  for  the  long  lines 
is  much  greater  than  for  the  short  lines.  For  an  example  of  the 
application  of  the  principle,  see  pp.  445-447  of  the  Appendix 
referred  to  in  the  preceding  section. 

This  method  of  determining  the  proper  weights  becomes  more 
reliable  the  more  extensive  the  net  and  the  greater  the  number 
of  lines  in  each  class. 

195.  It  sometimes  happens,  as,  for  example,  in  the  adjustment 
of  the  precise  level  net  covering  the  eastern  half  of  the  United 
States,  that  the  elevations  are  all  referred  to  mean  sea  level  by 
tidal  observations  connected  directly  with  the  net  at  various 
points.  In  this  case  the  method  of  adjustment  is  the  same  as 
outlined  above,  sea  level  at  atiy  point  being  treated  as  if  it  were 
the  one  bench-mark  in  the  above  statement  to  which  all  levels 
are  referred.* 

*  For  examples  of  complicated  level  net  adjustments,  see  Appendix  8, 
Coast  and  Geodetic  Survey  Report  for  1899,  and  Appendix  3  of  1903. 


APPLICATION    TO    LEVELING  271 

196.  The  adjustment  of  a  net  of  trigonometric  levels  does 
not  differ  essentially  from  that  outlined  above.  In  this  case, 
from  measurements  of  vertical  angles  at  various  triangulation 
stations,  combined  with  the  horizontal  distances  between  these 
stations  as  determined  by  the  triangulation,  the  differences  of 
elevations  between  the  inter-visible  stations  taken  in  pairs,  are 
computed.  These  computed  differences  of  elevation  are  treated 
as  the  direct  results  of  the  observation,  and  the  relative  elevations 
of  the  various  stations  are  derived  from  an  adjustment  of  the 
net. 


CHAPTER   IX 

APPLICATION   TO   THE    SELECTION    OF    METHODS    OF    OBSERVATION 

197.  The  preceding  chapters  have  dealt  with  the  principles 
of  least  squares  as  furnishing  a  method  of  computing  the  most 
probable  results  from  given  observations.  To  confine  the 
attention  entirely  to  this  application,  as  is  frequently  done,  even 
by  persons  thoroughly  familiar  with  the  principles,  is  to  over- 
look a  still  more  important  though  less  extensive  use  to  which 
these  principles  may  be  applied,  namely,  to  the  selection  of 
methods  of  observation. 

When,  in  any  class  of  precise  measurement,  the  method  of 
observation  is  selected,  the  maximum  accuracy  attainable  is 
thereby  fixed.  The  computer  cannot  improve  upon  what  the 
observer  has  done,  he  can  but  bring  out  all  its  excellence.  The 
observer  may,  by  improvements  in  his  method  of  observation, 
not  only  raise  the  standard  of  maximum  accuracy  attainable  in 
his  own  observations,  but  by  example  raise  it  for  all  later  ob- 
servers. Or  he  may  for  himself  and  later  observers  greatly 
reduce  the  amount  of  observing  necessary  to  attain  a  given 
standard  of  accuracy,  and  thereby  greatly  reduce  the  cost  of 
the  work.  Improvements  in  methods  of  observation  are  far- 
reaching  in  their  effects,  and  the  application  of  least  squares  to 
this  end  correspondingly  important. 

The  expression,  "selection  of  methods  of  observation,"  is 
here  used  in  a  general  sense  such  as  to  include  the  selection 
of  the  instrument  and  its  mounting  and  protection,  and  the 
selection  of  conditions  under  which  to  observe,  as  well  as  the 
mere  selection  of  a  method  of  manipulation  and  a  program  of 
observation. 

198.    A  clear  understanding  of  the  relative   influence  upon 

272 


SELECTION    OF    METHODS    OF    OBSERVATION         273 

the  results  of  errors  arising  from  various  sources  is  a  prime 
requisite  on  the  part  of  one  who  proposes  to  improve  methods 
of  observation.  It  is  also  of  great  importance  to  the  observer 
attempting  to  secure  a  maximum  of  accuracy  with  a  minimum 
of  expenditure.  A  working  knowledge  of  the  principles  of  least 
squares  is  essential  to  this  clear  understanding. 

In  general,  methods  of  observation  are  to  be  improved  : 

1.  By  reducing  at  their  source" the  errors  which  have  pre- 
dominating influence. 

2.  By  transferring  the  errors  from  any  given  source  from 
the  systematic  or  constant  class  into  the  accidental  class,  by  a 
change  in  instrument  or  method. 

3.  By  introducing  such  simplifications  in  instruments  and 
methods  as  will  increase  the  rapidity  of  observing,  possibly  at 
the  expense  of  making  slight  increases  in  such  errors  as  have 
little  influence  on  the  final  results.  In  general,  the  third  sug- 
gestion is  important  not  simply  for  economic  reasons,  but  also 
because  any  increase  in  the  rapidity  of  observing  is  likely  to 
lead  indirectly  to  an  increase  in  accuracy  by  reducing  instru- 
mental errors. 

A  treatment  of  the  application  of  the  principles  of  least 
squares  to  the  selection  of  methods  of  observation  within  the 
limits  of  a  single  chapter  must  be  suggestive  rather  than  com- 
plete. In  the  examples  given  to  illustrate  this  application,  con- 
siderable knowledge  of  instrument  and  method  of  observation 
will  be  assumed  to  be  possessed  on  the  part  of  the  reader. 
Attention  is  invited  to  the  principles  sketched  rather  than  to 
the  particular  numerical  estimates  of  magnitude  of  each  class 
of  errors  in  the  c\ami)]es. 

199.  Distinction  Between  Accidental,  Systematic,  and  Con- 
stant Errors.  —  In  discussing  errors,  and  especially  when  dis- 
cussing them  with  reference  to  their  ultimate  effects,  it  is  quite 
important  to  keep  clearly  in  mind  the  distinctions  between 
accidental  errors,  constant  errors,  and  systematic  errors.  A 
constant  error  is  one  wliirh   has    the  .same   effect   upon  all  llic 


274  THE    ADJUSTMENT    OF    OBSERVATIONS 

observations  of  the  series  or  portion  of  a  series  under  consider- 
ation. A  systematic  error  is  one  of  which  the  algebraic  sign, 
and,  to  a  certain  extent,  the  magnitude,  bears  a  fixed  relation 
to  some  condition  or  set  of  conditions.  Accidental  errors  are 
not  constant  from  observation  to  observation,  they  are  as  apt 
to  be  minus  as  plus,  and  they  presumably  follow  the  law  of 
error  which  is  the  basis  of  the  theory  of  least  squares.  Thus, 
for  example,  the  phase  error  in  observations  of  horizontal  direc- 
tions is  systematic  with  respect  to  the  azimuth  of  the  sun  and 
of  the  line  of  sight.  The  personal  equation  of  an  observer 
introduces  a  constant  error  into  the  observations  of  the  sepa- 
rate stars  in  a  time  set.  The  expression  "  constant  error  "  is 
often  used  loosely  in  contradistinction  to  "accidental  error," 
in  such  a  way  as  to  include  both  strictly  constant  errors  and 
systematic  errors. 

The  effect  of  accidental  errors  upon  the  final  result  may  be 
diminished  by  continued  repetition  of  the  observations  and  by 
the  least-square  method  of  computation.  The  effects  of  con- 
stant errors  and  of  systematic  errors  must  be  eliminated  by 
other  processes,  for  example,  by  changing  the  method  or  pro- 
gram of  observations,  by  special  investigations  or  by  special 
observations  designed  to  evaluate  a  constant  error  or  to  deter- 
mine the  exact  law  of  a  systematic  error. 

200.  More  Accurate  Definition  of  Probable  Error.  —  It 
cannot  be  emphasized  too  much  or  too  frequently  that  the 
theory  of  least  squares  applies  to  accidental  errors  only,  that 
the  least-square  method  of  computation  is  designed  to  secure 
efficient  elimination  of  the  effects  of  accidental  errors,  but  may 
or  may  not  be  efficient  in  reducing  the  effects  of  errors  of  other 
kinds,  and  that  a  probable  error  derived  directly  from  residuals 
is  an  adequate  measure  of  accidental  errors  only.  The  most 
frequent,  and  perhaps  the  most  serious,  mistakes  made  in 
applying  the  theory  of  least  squares  arise  from  disregarding 
the  points  here  emphasized. 

The  definition  of  probable  error  as  ordinarily  given  is  equiva- 


SELECTION    OF    METHODS    OF    OBSERVATION  275 

lent  to  saying  that  the  chances  are  even  for  or  against  the 
proposition  that  a  certain  stated  value  having  given  a  probable 
error  does  not  differ  from  the  truth  by  more  than  that  probable 
error.  That  is,  if  the  probable  error  of  a  single  observation 
of  an  angle  is  stated  to  be  ±  i.oo"  it  is  understood  that  the 
chances  are  even  for  and  against  the  proposition  that  the  result 
of  any  one  observ^ation  is  within  i.oo''  of  the  truth. 

If  the  probable  error  in  question  is  one  which  is  based 
directly  upon  residuals,  this  definition  should  be  modified  to 
limit  it  so  as  to  refer  to  the  accidental  errors  only.  This  may 
be  done  by  stating  that  in  the  above  case  the  chances  are  even 
for  or  against  the  proposition  that  the  result  of  any  one  obser- 
vation is  within  i.oo"  of  the  mean  value  which  would  result 
from  an  infinite  number  of  such  observations  made  under  the 
same  average  conditions  as  the  observation  in  question.  This 
form  of  definition  is  non-committal  as  to  possible  constant  errors 
affecting  all  the  observations,  and  as  to  systematic  errors  which 
might  change  in  magnitude  and  algebraic  sign,  if  there  were 
changes  in  the  conditions.  Such  a  definition  differs  from  the 
ordinary  definition  in  stating  that  the  probable  error  is  a  meas- 
ure of  the  departure  to  be  expected  from  a  mean  of  an  infinite 
number  of  such  results  as  are  being  considered,  rather  than 
from  the  truth.  The  suj^posed  mean  of  an  infinite  number  of 
results  would  be  free  from  the  effects  of  accidental  error,  would 
be  as  much  affected  by  constant  error  as  any  one  observation, 
and  would  be  less  subject  to  systematic  error  only  to  the  degree 
determined  by  the  frequency  and  extent  to  which  the  conditions 
were  allowed  to  vary. 

There  is  no  sharply  defined  line  of  separation  between  acci- 
dental and  systematic  errors.  The  perfect  type  of  accidental 
error,  the  type  upon  which  the  theory  of  least  squares  is  based, 
is  an  error*  which  is  the  algebraic  sum  of  an  infinite  number  of 
independent  infinitesimal  elemental  errors,  all  ccjual  in  magni- 
tude, and  each  as  likely  in   any  given   case  to  be   positive  as 

*  See  p.  161. 


276  THE    ADJUSTMENT    OF    OBSERVATIONS 

negative.  In  any  actual  case  the  number  of  elemental  errors 
is  finite,  each  of  them  is  finite  in  magnitude,  and  the  elemental 
errors  due  to  different  causes  may  be  of  very  different  average 
magnitude.  Each  elemental  error  actually  depends  both  in 
magnitude  and  sign  upon  certain  conditions,  though  those  con- 
ditions may  be  unknown,  and  is,  therefore,  a  systematic  error. 
The  conditions  upon  which  different  elemental  errors  depend 
may  not  be  independent.  Indirect  evidence  shows  nevertheless 
that  in  many  cases  the  errors  of  observation  were  sensibly  of 
the  perfectly  accidental  type,  that  is,  the  relation  between  the 
numbers  of  errors  of  various  magnitudes  and  signs  is  sensibly 
that  which  must  hold  for  the  perfect  type  of  accidental  error 
(see  p.  31  and  Table  I),  and  therefore  all  deductions  based 
upon  this  law  of  error  are  sensibly  true.  In  other  contrasting 
cases,  certain  of  the  elemental  errors  may  be  easily  proved  to  be 
in  the  systematic  class,  and  may  be  segregated  from  the  re- 
maining elemental  errors  which  then,  as  combined,  belong 
sensibly  to  the  accidental  class.  In  the  intermediate  and  most 
frequent  cases,  there  are  slight  observable  departures  from  the 
law  of  error  corresponding  to  the  perfect  type  of  accidental 
error,  and  the  deductions  based  upon  that  law  are  slightly,  but 
observably,  at  variance  with  the  facts.  In  such  a  case  some 
systematic  error  exists  which  is  almost,  but  not  quite,  detected, 
and  one  is  forced  to  treat  all  the  errors  as  being  of  the  acci- 
dental class. 

201.  Detecting  Systematic  or  Consatnt  Errors.  —  It  is  im- 
portant that  the  computer  should  detect  systematic  or  constant 
errors  in  order  that  he  may  modify  his  method  of  computation 
to  correspond.  It  is  still  more  important  that  the  investigator 
of  methods  of  observation  should  detect  systematic  and  constant 
errors  in  order  that  they  may  thereafter  be  avoided.  It  is 
much  more  important  to  avoid,  or  to  provide  special  means  of 
eliminating,  systematic  or  constant  errors  than  accidental  errors, 
for  the  reason  that  accidental  errors  are  rapidly  eliminated  by 
the  mere  process  of  increasing  the  number  of  observations. 


SELECTION    OF    METHODS    OF    OBSERVATION  277 

A  considerable  variety  of  methods  may  be  used  for  the 
detection  of  systematic  and  constant  errors.  Each  method  will 
be  found  as  a  rule  to  correspond  to  one  or  another  of  the  fol- 
lowing five  cases: 

Case  i.  Systematic  errors  may  sometimes  be  detected  by 
noting  a  tendency  of  the  residuals  to  have  a  certain  algebraic 
sign  when  certain  conditions  exist,  and  the  opposite  sign  when 
these  conditions  are  absent  or  the  opposite  conditions  occur. 

Case  2.  Errors  which  are  constant  in  each  of  a  number  of 
groups  of  observations  may  sometimes  be  detected  by  noting 
that  the  disagreements  between  the  mean  results  for  each  group 
are  greater  than  can  be  accounted  for  by  the  probable  errors  of 
those  means  as  computed  from  the  residuals  within  each  group. 
Or,  what  is  equivalent,  errors  which  are  constant  for  each  of  a 
number  of  groups  of  observations  may  sometimes  be  detected 
by  noting  that  the  probable  error  of  the  mean  of  all  the  obser- 
vations is  apparently  much  larger  if  computed  from  the  residuals 
of  the  mean  of  each  group  from  the  mean  of  all,  than  if  computed 
from  the  residuals  of  each  observation  from  the  mean  of  all. 
When  it  is  recognized  that  there  are  errors  which  are  constant  for 
each  group,  these  errors  may  in  some  cases  be  proved  to  belong 
to  the  systematic  class  by  detecting  a  relation  between  the  errors 
peculiar  to  each  group  and  some  condition  peculiar  to  that  group. 

Case  3.  Systematic  errors  may  sometimes  be  detected  by 
comparing  the  relative  frequency  of  residuals  of  different  mag- 
nitude and  sign  with  the  theoretical  relation,  according  to  the 
law  of  error,  between  the  magnitude  and  the  frequency  of  er- 
rors. This  comparison  may  be  made  by  using  Table  I,  or  by 
plotting  the  actual  curve  of  the  residuals  with  the  theoretical 
curve  of  error  superposed  upon  it.  The  comparison  may  show 
that  there  are  relatively  many  more  very  large  residuals  than 
would  be  the  case  if  the  errors  were  all  accidental.  The  exami- 
nation of  the  conditions  corresponding  to  these  large  residuals 
may  then  lead  to  the  detection  of  the  law  and  cause  of  these 
large  systematic  errors. 


278  THE    ADJUSTMENT    OP    OBSERVATIONS 

Case  4.  That  systematic  or  constant  errors  may  arise  from 
a  given  source  may  sometimes  be  proved  by  special  observations 
for  that  purpose. 

Case  5.  Either  systematic  or  constant  errors  may  sometimes 
be  detected  by  comparing  the  results  of  the  observations  in 
question  with  results  obtained  independently  from  observations 
of  an  entirely  different  kind. 

The  detection  of  systematic  or  constant  errors  necessarily 
involves  least  squares  as  a  basis,  but  this  must  be  supplemented 
by  something  else,  as  the  method  of  least  squares  deals  with 
accidental  errors  only. 

Examples  of  each  of  these  cases  will  be  found  in  the  text 
which  follows : 

202.  Zenith  Telescope  Latitude  Observations.  —  Observa- 
tions with  a  zenith  telescope  for  latitude  are  especially  interest- 
ing as  illustrating  errors  which  are  a  close  approximation  to  the 
perfect  type  of  accidental  error.  The  zenith  telescope  and  in- 
strument is  an  example  of  proper  selection.  It  is  difficult  to  im- 
prove upon  it  for  the  reason  that  the  errors  from  every  important 
source  are  of  the  accidental  class,  and  the  effects  of  the  errors 
arising  from  various  sources  upon  the  final  results  are  so  nearly 
of  the  same  magnitude  that  little  gain  in  accuracy  may  be  se- 
cured except  by  reducing  the  errors  from  several  sources. 

As  indicating  how  the  error  in  a  result  from  an  observation 
of  a  single  pair  of  stars  with  a  zenith  telescope  approaches  the 
ideal  accidental  error,  which  is  supposed  to  be  the  algebraic 
sum  of  an  infinite  number  of  independent  infinitesimal  errors, 
all  equal  in  magnitude,  and  each  as  likely  in  any  case  to  be 
positive  as  negative,  sixteen  independent  elemental  errors  in 
this  result  may  be  named,  each  capable  of  introducing  an 
accidental  error  of  from  ±0.01"  to  ±0.16''  into  the  result,  of 
which  the  probable  error  due  to  all  causes  is  iL  0.20"  to  rt  0.30". 
Several  of  the  sixteen  errors  named  could  if  desired  be  separated 
by  more  minute  examination  into  other  elemental  errors  as 
suggested  in  the  text,  so  that  the  number  of  elemental  errors  is 


SELECTION    OF    METHODS    OF    OBSERVATION  279 

really  several  times  sixteen.     The  list  of  elemental  errors  as 
given  is  suggestive  rather  than  complete. 

203.  Elemental  Errors  in  Zenith  Telescope  Observations. 
—  I.  The  error  of  bisection  of  the  star  image  by  the  microm- 
eter line,  depending  among  other  things  upon  the  observer's 
perception  and  his  control  of  the  muscles  of  his  fingers,  the 
shape  of  the  image,  the  defects  of  the  observer's  eyes,  the 
irregular  motion  of  the  image  due  to  momentary  changes  in 
refraction,  irregularities  in  the  line  used  in  making  the  bisection, 
the  magnitude  of  the  star  and  the  lighting  of  the  field  of  view 
and  the  line. 

2.  The  error  of  reading  the  position  of  the  bubble  in  the 
level,  depending  upon  the  lighting  and  upon  parallax  as  well  as 
upon  the  observer's  estimate  of  tenths  of  a  division  and  his 
perception. 

3.  The  error  caused  by  unequal  heating  of  the  level  vial  and 
consequent  movements  of  the  bubble. 

4.  The  error  caused  by  the  error  in  the  assumed  value  of 
one  division  of  the  level.  The  error  in  the  value  of  a  division 
is  due  in  part  to  errors  in  determining  it  and  in  part  to  variation 
of  the  actual  value  from  time  to  time. 

5.  The  error  of  reading  the  micrometer  head. 

6.  The  error  due  to  the  error  in  the  assumed  value  of  one 
turn  of  the  screw.  The  remark  made  in  connection  with  the 
level  value  applies  here  also. 

7.  The  error  due  to  non-uniformity  of  the  screw  throughout 
its  length. 

8.  The  error  due  to  periodical  errors,  having  a  period  of  one 
turn  in  the  screw  and  its  nut. 

9.  The  error  due  to  the  inclination  of  the  bisecting  line  to 
the  horizon  and  to  the  difficulty  of  making  all  bisections  on 
exactly  the  same  part  of  the  line. 

10.  The  error  due  to  inclination  of  the  horizontal  axis  of  the 
instrument. 

1 1.  The  error  due  to  the  azimuth  error  of  the  instrument. 


28o  THE    ADJUSTMENT    OF    OBSERVATIONS 

12.  The  error  due  to  the  collimation  error  of  the  instrument. 

13.  The  error  due  to  variations  in  the  angle  between  the 
tangent  to  the  level  vial  at  its  middle  point  and  the  hne  of  col- 
limation, this  variation  being  due  to  changes  in  temperature  in 
different  parts  of  the  instrument  as  well  as  to  stresses. 

14.  The  error  due  to  variation  of  the  differential  refraction 
from  its  assumed  mean  value.  This  variation  is  dependent  upon 
the  conditions  as  to  temperature  and  pressure  at  all  points  along 
many  miles  of  each  of  the  two  lines  of  sight. 

15.  The  error  due  to  the  error  in  the  declination  of  each  star 
as  used  in  the  computation.  The  error  of  declination  is  due  in 
general"  to  the  separate  errors  in  dozens  or  perhaps  even  hun- 
dreds of  observations  made  at  various  times  at  many  different 
observatories,  each  observation  being  in  general  affected  by  as 
many  elemental  errors  as  are  suggested  in  the  preceding  fifteen 
numbers  of  this  list. 

In  estimating  the  number  of  elemental  errors  affecting  a 
result  which  are  represented  by  this  list,  it  should  be  noted  that 
each  result  depends  upon  the  observations  on  two  stars. 

204.  In  determining  the  latitude  of  a  station  by  zenith  tele- 
scope method,  a  program  frequently  followed  is  to  observe  sev- 
eral pairs  of  stars,  say  twenty,  on  each  of  several  nights,  five, 
for  example.  The  probable  error  of  a  single  observation  is  com- 
puted from  the  residuals  of  each  observation  from  the  mean  of 
the  five  observations  on  that  pair.  When  the  mean  results  for 
the  different  pairs  are  compared,  they  are  found  to  show  dis- 
agreements which  are  greater  than  can  be  accounted  for  by  the 
residuals  within  each  group.  This  test,  which  is  an  illustration 
of  Case  2,  shows  that  there  is  some  error  in  the  results  which  is 
constant  for  each  group.  In  this  case  it  seems  obvious  that  this 
constant  error  in  each  group  is,  in  the  main,  simply  the  error  in 
the  mean  of  the  two  declinations  of  the  stars  of  the  pair.  This 
declination  error,  which  is  constant  for  each  pair,  belongs  mainly 
in  the  accidental  class  when  the  results  from  various  pairs  are 
considered. 


SELECTION    OF    METHODS    OP    OBSERVATION  281 

There  may  be  other  errors  which  are  constant  for  each  pair 
and  which  combine  with  these  dechnation  errors ;  for  example, 
errors  No.  6  and  7,  and  some  parts  of  No.  i  of  the  above  list. 
For  the  present  purpose  it  is  not  sufficient  to  stop  with  the 
obvious  conclusion  that  all  or  nearly  all  of  the  error  which  is 
constant  for  each  pair  is  due  to  declinations.  It  is  desirable,  if 
possible,  to  prove  it.  Two  lines  of  proof  are  available  and  have 
been  used. 

Let  e^  be  the  probable  error  of  the  mean  result  from  a  single 
pair  as  derived  from  the  residuals  of  these  various  mean  results 
from  the  mean  of  all  for  the  station.  Let  e  be  the  average 
probable  error  of  a  mean  result  for  a  pair  as  derived  from  the 
residuals  of  the  separate  observations  on  that  pair  from  the 
mean  result  for  that  pair.  Let  e*j,  be  the  average  probable 
error  of  the  mean  of  two  star  declinations  for  the  particular  stars 
observed.  The  apparently  obvious  assumption  suggested  in  the 
preceding  two  paragraphs  is  expressed  by  the  formula:* 

<?«  is  the  only  unknown,  and  its  value  may  therefore  be  computed 
from  the  latitude  observations.  Its  value  may  also  be  derived 
from  the  computations  made  by  the  astronomer  in  combining 
the  observ^ations  at  various  observatories  and  at  various  times. 
These  two  computations  give  usually  nearly  the  same  values 
for  e**  the  value  from  the  latitude  observations  being,  uj^on 
the  average,  very  slightly  larger  than  that  computed  by 
the  astronomer.  Hence,  the  errors  which  are  constant  for 
each  pair  are  nearly,  but  not  quite,  all  due  to  errors  of 
declinations. 

If  several  latitude  stations  along  the  same  parallel  have  been 
occupied  and  the  same  list  of  pairs  used  at  all  stations,  it  becomes 
possible  to  apply  in  a  slightly  different  way  the  principles 
involved  in  the  preceding  paragraph.     This  was  done  for  twelve 

*  See  Appendix  7  of  the  Coast  and  Geodetic  Survey  Report  for  iSgS, 
Longitude,  Latitude,  Azimutli,  p.  358. 


282  THE    ADJUSTMENT    OF    OBSERVATIONS 

stations  along  the  Mexican  boundary.      It  confirmed  the  con- 
clusion of  the  preceding  paragraph.* 

205.  Let  it  be  supposed  that  the  probable  error  of  the  mean 
of  the  two  declinations,  e**,  is  zl::0.i6",  and  that  the  probable 
error  of  a  single  observation  of  a  pair  is  ±0.30".  The  first  of 
these  probable  errors  represents  the  accuracy  to  be  expected  in 
the  declinations  which  are  now  available.  The  second  is  easily 
attained  with  a  good  portable  zenith  telescope.  Under  these 
conditions,  which  is  the  better  program,  to  observe  20  pairs 
on  each  of  five  nights,  100  observations  in  all,  five  on  each  pair, 
or  to  observe  100  pairs  each  once  only,  the  observations  being 
scattered  over  as  many  nights  as  may  be  necessary  to  secure 
them.?  If  20  pairs  are  observed  on  each  of  five  nights,  the 
probable  error  of  the  final  result  will  be, 


v/' 


^7^  +  -^  =  ^(.036)^  +  (.030)2  =  ±  .047. 
20         100 

If   100  pairs  are  each  observed  once,  the  probable  error  of  the 
final  result  will  be 


-^  H V(.oi6)-  +  (.030)-  =  ±  .034, 

100       100 


showing  a  very  decided  advantage  in  favor  of  this  program  of 
observation.  The  advantage  becomes  still  more  evident  when 
it  is  noted  that  if  54  pairs  be  each  observed  once,  the  probable 
error  of  the  final  result  will  be 


V/: 


the  same  as  would  be  obtained  from  100  observations  on  20 
pairs,  but  secured  with  little  more  than  one-half  as  much  observ- 
ing.    It  would  evidently  be   a   great   improvement  to  change 

*  Report  of  the  Boundary  Commission  upon  the  Survey  and  Re-marking 
of  the  Boundary  between  the  United  States  and  Mexico,  West  of  the  Rio 
Grande,  1891-1896,  p.  105. 


SELECTION    OF    METHODS    OF    OBSERVATION  283 

from  the  plan  usually  followed  in  the  place  of  observing  each 
pair  ioin-  or  more  times  to  the  plan  of  observing  each  pair  but 
once.  This  improvement  is  one  which  almost  inevitably  suggests 
itself  to  a  person  looking  at  the  subject  from  a  least-square 
point  of  view,  whereas  the  writer's  experience  indicates  that 
those  who  do  not  take  this  view-point  fail  to  appreciate  the 
desirability  of  this  improvement.  For  an  example  of  the  com- 
parative results  by  the  two  plans,  see  the  Mexican  Boundary 
Report  referred  to  on  the  preceding  page,  pp.  106,  107.  At 
the  typical  station  represented  by  that  series  of  observations, 
108  observations  on  72  pairs  gave  a  much  more  accurate  result 
than  could  have  been  obtained  even  from  an  infinite  number  of 
observations  on  18  pairs. 

206.  Telegraphic  Longitude  Observations.  —  Determinations 
of  differences  of  longitude  by  tlie  telegraphic  method  furnish 
illustrations  of  the  detection  of  systematic  or  constant  errors 
by  the  methods  of  Case  i  and  Case  4  and  the  following  three 
illustrations  of  Case  2. 

1.  For  the  transits  ordinarily  used  in  telegraphic  longitude 
determinations  in  the  Coast  and  Geodetic  Survey,  the  probable 
error  of  an  observing  transit  of  a  star  over  a  single  line  is 
usually  less  than  ±0.10",  as  computed  from  the  residuals  of  the 
separate  lines  from  the  mean  of  the  11  lines  on  which  obser- 
vations were  taken.*  On  this  basi.s,  the  probable  error  of  the 
transit  across  the  mean  of  the  1 1  lines  would  be  less  than 
±0.03".  This  same  probable  error,  as  computed  fr()m  the 
residuals  of  the  separate  stars  from  the  mean  for  the  time  set, 
is  usually  considerably  larger,  say  zt  0.04''  upon  an  average. 
This  indicates  that  the  errors  of  the  observations  would  not  be 
much  reduced  by  increasing  the  number  of  lines,  say  from 
II  to  21. 

2.  From  fifteen  Coast  and  Geodetic  Survey  longitude  determi- 

*  The  numerical  estimates  of  errors  used  in  tiiis  chapter  are  taken,  as  a 
rule,  from  Appendix  7  of  the  Coast  and  Geodetic  Survey  Report  for  i8y8, 
Time,  Longitude,  Latitude,  and  Azimuth, 


2S4  THE    ADJUSTMENT    OF    OBSERVATIONS 

nations  involved  in  the  primary  longitude  net,  it  was  found  that 
whereas  the  probable  error  of  a  difference  of  longitude  from  one 
night's  observation  as  computed  from  the  residuals  of  different 
stars  from  the  mean  for  the  night  was  ±  0.013",  if  this  prob- 
able error  were  computed  from  the  residuals  of  different  nights 
from  the  mean  of  the  ten  nights  concerned  in  the  determination 
(the  correction  for  relative  personal  equation  being  already 
applied),  it  was  enough  larger  to  indicate  that  the  constant  error 
peculiar  to  each  night  was  i  0.022.  Hence  there  would  be 
little  appreciable  gain  in  accuracy  if  the  number  of  observa- 
tions per  night  were  greatly  increased. 

3.  From  many  longitude  determinations  involved  in  the  pri- 
mary longitude  net,  and  each  consisting  as  a  rule  of  ten  nights 
of  observation,  it  was  found  that  whereas  the  probable  error  of 
the  mean  of  the  ten  nights  as  computed  from  the  residuals 
of  the  separate  nights  from  the  mean  was  zt  0.0 1 1",  the  probable 
error  as  computed  from  the  adjustment  of  the  net  was  so  much 
larger  as  to  show  that  there  was  a  constant  error  peculiar  to 
each  mean  of  ten  nights  of  zt  0.022".  It  follows  that  a  reduc- 
tion in  the  number  of  nights  to  six  or  even  four  would  result  in 
but  a  slight  increase  in  accuracy,  —  say  10  per  cent. 

In  the  longitude  determinations  referred  to  above,  the  usual 
procedure  was  for  the  two  observers  to  exchange  places  after 
the  first  five  of  the  ten  nights  of  observation.  In  all  other 
respects  except  this,  the  last  five  nights  of  observation  were 
made  under  conditions  as  nearly  as  possible  identical  with  those 
during  the  first  five  nights.  If  the  mean  of  the  ten  nights  is 
taken,  and  the  corresponding  residuals  written  out,  it  is  evident, 
as  a  rule,  that  there  is  a  tendency  for  the  residuals  of  one  group 
of  five  to  have  one  sign,  and  of  the  other  group  to  have  the 
opposite  sign.  This  is  an  illustration  of  Case  i,  and  indicates 
that  there  is  a  systematic  error  in  the  result  which  bears  a  fixed 
relation  to  the  relative  position  of  the  observers,  and  is  there- 
fore due  to  their  relative  personal  equation.  Accordingly  it  is 
eliminated  by  taking  one-half  the  difference  of  the  two  groups 


SELECTION    OF    METHODS    OF    OBSERVATION  285 

of  five  as  the  relative  personal  equation,  correcting  each  night's 
result  by  this  amount,  and  deriving  the  final  difference  of  longi- 
tude from  these  corrected  values.  A  confirmation  of  the  sup- 
position that  this  systematic  error  is  the  relative  personal  equa- 
tion, is  furnished  by  comparing  successive  values  of  the  relative 
personal  equation  as  thus  derived  for  the  same  pair  of  observers 
in  successive  longitude  determinations. 

In  connection  with  some  longitude  determinations,  the  rela- 
tive personal  equation  of  the  two  observers  has  been  determined 
by  a  personal  equation  machine,  or  by  observations  by  the 
half-transit  method.  Either  of  these  is  an  illustration  of  Case  4. 
Similarly  the  method  of  Case  4  has  been  applied  to  prove  that 
the  systematic  error  in  a  longitude  determination  arising  from 
the  action  of  the  single  relays  which  connect  the  mean  tele- 
graphic line  at  each  longitude  station  with  the  chronograph 
circuit  must  be  in  the  thousandths  of  seconds  only,  not  in  hun- 
dredths. This  was  done  by  making  special  observations  of 
changes  which  occur  in  the  time  of  operation  of  these  relays 
when  extreme  changes  are  made  in  their  adjustment  and  in 
the  strength  of  the  currents  operating  them. 

A  study  of  the  sources  of  error  in  telegraphic  longitude 
determinations  by  the  methods  suggested  in  this  chapter,  but 
necessarily  in  much  greater  detail  than  it  is  possible  to  give 
here,  inevitably  leads  to  the  conclusion  that  a  considerable 
portion  of  errors  which  are  constant  for  a  night  are  due  to 
variation  of  the  relative  personal  equation  of  the  two  observers. 
This  suggests  that  the  line  along  which  some  improvements  in 
the  methods  should  be  sought  is  that  of  securing  some- means 
of  making  the  relative  personal  equation  and  its  variation  zero. 
For  this  purpose  a  new  attachment  to  the  astronomical  transit, 
known  as  a  transit  micrometer,  was  put  into  use  several  years 
ago  by  the  Prussian  Geodetic  Institute  with  great  success,  and 
is  now  in  use  in  this  country.* 

*  See  Appendix  S  of  the  Coast  and  Geodetic  Survey  Report  for  1904, 
The  Transit  Micrometer. 


286  THE   ADJUSTMENT    OF    OBSERVATIONS 

207.  Other  Illustrations.  —  As  another  illustration  of  Case 
2,  it  may  be  noted  that  in  the  measurement  of  angles  in  primary 
triangulation  the  general  experience  is  that  the  probable  error 
of  an  angle  as  computed  from  the  residuals  of  the  various  meas- 
ures of  that  angle  from  their  mean,  or  even  computed  from  the 
residuals  which  occur  in  a  local  adjustment  involving  all  the 
angles  at  a  station,  is  very  much  smal'^r  than  the  probable 
error  of  an  angle  computed  from  the  figure  adjustment,  which 
involves  a  much  larger  group  of  observations.  This  indicates 
that  there  are  errors  which  are  constant  for  a  station  and  are 
not  affected  by  either  increasing  or  decreasing  the  number  of 
measurements  of  an  angle.  Acting  upon  this  reasoning,  the 
Coast  and  Geodetic  Survey  has  recently  reduced  the  number 
of  observations  of  each  angle  in  primary  triangulation  from  22 
to  34,  to  16.  This  very  important  saving  in  time  and  money 
has  not  been  accompanied  by  any  appreciable  decrease  in 
accuracy. 

208.  An  examination  of  observations  of  astronomical  azimuth 
at  many  stations  has  shown  that  frequently  the  residuals  for 
each  night  tend  to  stand  in  a  group  by  themselves,  all  having 
one  sign.  This  examination  is  one  form  of  application  of  the 
method  of  Case  2.  It  indicates  that  there  are  constant  errors, 
in  some  instances,  at  least,  peculiar  to  each  night  in  azimuth 
observations,  and  that  therefore,  if  the  highest  degree  of  accu- 
racy is  desired,  the  observations  must  be  extended  over  several 
nights.  The  same  test  applied  to  latitude  observations  made 
with  a  zenith  telescope,  indicates  in  general  that  there  are  no 
constant  errors  peculiar  to  each  night,  though  to  this  statement 
some  exceptions  have  been  noted. 

209.  Two  interesting  illustrations  of  Case  2  are  furnished  by 
the  use  of  a  five-meter  iced  bar  in  the  standardization  of  base 
apparatus  in  the  Coast  and  Geodetic  Survey.  In  determining 
the  length  of  the  bar,  it  was  found  that  the  residuals  apparently 
indicated  that  the  bar  was  2  to  4  microns  longer  when  its  length 
was  determined  while  its  A-end  lay  to  the  left  of  the  observer 


SELECTION    OF    METHODS    OF    OBSERVATION  287 

than  when  it  lay  m  the  reverse  position.*  Though  this  is  a 
very  small  quantity,  it  was  persistently  shown  by  the  observa- 
tions. It  was  recognized,  when  it  had  been  carefully  studied,  as 
a  systematic  error  due  to  the  personal  equation  of  the  observers 
in  making  bisections  of  the  graduations  on  the  bar.  To  secure 
the  highest  possible  degi-ee  of  accuracy  in  determining  the 
length  of  the  bar  and  in  using  it  as  a  standard,  it  is  therefore 
necessary  to  make  one-half  the  observations  with  the  bar  in 
each  position  and  therefore  eliminate  this  systematic  error. 

Again,  in  using  this  iced  bar  in  the  open  air  to  measure  a 
standard  length  of  100  meters,  it  was  found  that  the  residuals 
from  the  mean  of  all  the  measures  was  of  one  sign  if  the  meas- 
uremicnt  had  progressed  toward  the  sun,  and  of  the  opposite 
sign  if  the  measurement  had  progressed  away  from  the  sun.f 
This  systematic  error  was  eliminated,  in  part  at  least,  by  making 
the  measurements  in  pairs  with  the  progress  in  opposite  direc- 
tions in  the  two  measurements  of  each  pair,  and  with  as  short 
an  intervening  time  interval  as  possible.  This  systematic  error 
is  believed  to  be  due  to  slight  motions  of  the  microscopes  due  to 
changes  of  temperature. 

210.  A  good  illustration  of  Case  3  is  furnished  by  a  com- 
parison of  observed  and  predicted  tides  at  Sandy  Hook,  N.J.J 
The  difference  between  the  observed  and  predicted  heights  of 
high  and  low  water  was  due  to  many  separate  elemental  errors, 
the  errors  in  the  six  years  of  tidal  observations  from  which  tlie 
tidal  constants  used  in  the  prediction  were  derived  and  in  the 
one  year  of  observations  with  which  the  comparison  was  made, 
the  errors  in  the  theory  involved  in  the  computation  of  the 
tidal  constants,  and  the  errors  in  the  operation  of  the  predicting 
machine  itself.      It   was  desired   to  determine  as   fully  as  pos- 

*  Appendix  8,  Coast  and  (Geodetic  Survey  Report  for  1S92,  the  Holton 
Base,  pp.  382-391. 

t  Appendix  3,  Coast  and  Geodetic  Survey  Report,  igoi,  the  Measurement 
of  Nine  liases,  p.  245. 

t  Appendix  15,  Coast  and  Geodetic  Survey  Report  for  1S90,  and  espe- 
cially illustrations  Nos.  66  and  67  of  that  Report. 


288  THE    ADJUSTMENT    OF    OBSERVATIONS 

sible  what  systematic  errors  existed,  the  magnitude  of  the  acci- 
dental errors,  and  especially  how  large  were  the  errors  due  to 
the  machine. 

There  were  four  groups  of  f  oo  each  to  be  considered,  arising 
respectively  from  the  prediction  of  high-water  heights,  low-water 
heights,  high-water  times  and  low-water  times.  In  addition 
to  other  tests  applied,  the  probable  error  of  a  single  prediction 
was  computed  for  each  of  these  groups  (after  the  constant  error 
had  been  removed),  the  curve  of  distribution  of  errors  for  each 
group  drawn  on  a  large  scale  (the  magnitudes  of  the  error  being 
the  abscissae,  and  the  number  of  such  errors  the  ordinates  of 
the  curve),  and  the  theoretical  law  of  error  (Art.  27)  drawn 
to  the  same  scale  was  superposed  on  it. 

For  the  two  curves  corresponding  to  predicted  heights  it  was 
at  once  apparent  that  there  was  a  systematic  difference  in  char- 
acter between  the  actual  and  theoretical  curves.  The  actual 
curve  in  each  case  showed  about  13  times  as  many  errors  as 
the  theoretical  curve  greater  than  4J  times  the  computed  prob- 
able error,  and  about  79  times  as  many  greater  than  5|  times 
the  computed  error.  It  also  showed  about  J  as  many  errors  as 
the  theoretical  curve  less  than  the  computed  probable  error, 
and  about  f  as  many  between  one  and  three  times  the  computed 
probable  error.  A  difference  of  this  kind  between  the  actual 
and  theoretical  curves  indicates  that  there  is,  in  addition  to  the 
actual  errors,  some  large  systematic  error  which  occurs  occa- 
sionally, in  this  case  about  one  in  twenty  times  upon  an  aver- 
age. The  unusual  number  of  large  residuals  would  be  caused 
directly  by  such  a  systematic  error.  An  indirect  effect  of 
these  large  residuals,  due  mainly  to  the  large  systematic  error 
superposed  on  the  accidental  errors,  would  be  to  make  the  com- 
puted probable  error  much  too  large.  This  in  turn  would  cause 
the  theoretical  curve  to  depart  from  the  actual  between  errors 
of  zero  and  three  times  the  probable  error,  in  a  manner  similar 
to  that  noted  above.  Following  up  the  conclusion  that  the 
particularly  large  residuals   indicated  by  the  outer  portions  of 


SELECTION    OF    iMETHODS    OF    OBSERVATION  289 

the  actual  curve  were  due  to  a  systematic  error,  considerable 
evidence  was  found  that  they  were  due  to  effects  of  storms 
upon  mean  sea  level.  As  the  principal  purpose  of  the  test  was 
to  determine  how  great  were  the  errors  due  to  the  action  of  the 
tide  predicting  machine,  this  conclusion  that  the  large  errors 
were  not  chargeable  to  the  machine  was  an  important  one. 

The  comparison  between  the  actual  and  theoretical  curves 
for  predicted  times  of  high  and  low  water  showed  a  very  close 
agreement.  Storms  are  known  to  have  but  little  effect  upon 
the  time  of  high  and  low  water,  hence  the  systematic  errors  due 
to  this  cause  should  be  expected  to  be  small,  as  the  curves 
indicated  them  to  be. 

211.  Trigonometrical  leveling,  that  is,  leveling  by  observations 
of  vertical  angles  taken  in  connection  with  triangulation, 
frequently  connects  points  that  are  also  connected  by  precise 
leveling,  and  thus  furnish  an  illustration  of  Case  5.  The  test 
applied  by  the  precise  level  usually,  but  not  always,  indicates 
that  the  systematic  and  constant  errors  in  theoretical  leveling 
are  so  small  as  to  be  almost  or  quite  concealed  by  the  accidental 
errors. 

212.  The  following  three  illustrations  of  Case  5  are  all 
taken  from  TJie  Solar  Parallax  and  Its  Related  Constants,  by 
William  Harkness. 

1.  The  aberration  constant  has  been  determined  many  times 
and  by  many  methods,  among  which  are :  a,  by  observations  of 
right  ascensions  of  stars  with  an  instrument  in  the  meridian; 
/;,  by  observations  of  the  declinations  of  stars  with  an  instrument 
in  the  meridian ;  c,  by  observations  with  an  instrument  in  the 
prime  vertical;  d,  by  zenith  telescope  observations.  A  com- 
parison of  the  results  by  the  various  methods  indicates  clearly 
that  they  arc,  as  a  rule,  sul)ject  to  constant  or  systematic  errors 
much  larger  than  the  uneHminated  effects  of  accidental  errors. 

2,  The  flattening  of  the  earth  has  l)een  deri\'cd:  a,  from 
geodetic  arcs;  b,  from  pendulum  observations;  c,  from  the 
observed  precession  and  nutation  ;  d,  from  perturbations  of  the 


290  THE    ADJUSTMENT    OF    OBSERVATIONS 

moon.  The  comparison  of  the  various  results  indicates  that 
the  systematic  or  constant  errors  in  some  if  not  all  of  them  are 
much  larger  than  the  uneliminated  effects  of  accidental  errors. 
3.  The  mean  density  of  the  earth  has  been  determined  :  a, 
by  observations  of  the  attraction  of  mountains  as  measured  by 
the  deviations  of  the  plumb-line  in  the  immediate  vicinity;  b,  by 
observing  the  attraction  of  known  masses  of  matter  either  with 
a  torsion  balance  or  a  pendulum;  c,  by  pendulum  observations 
at  different  distances  from  the  center  of  the  earth,  near  mean 
sea  level  and  on  mountain  tops,  or  at  the  surface  of  the  earth 
and  in  mines ;  d,  by  observations  with  balances  of  the  ordinary 
form  either  of  the  attraction  of  a  known  mass  or  of  the  change 
in  the  attraction  of  the  earth  upon  a  known  mass  when  it  is 
moved  to  a  higher  or  a  lower  position.  The  comparison  shows 
systematic  or  constant  errors  in  the  results  which  are  large  in 
comparison  with  the  uneliminated  accidental  errors,  as  a  rule, 
when  methods  a  and  c  are  used,  and  in  some  cases  even  when 
other  methods  are  used. 


APPENDIX 


213. 


Values  of  0 


2      /"p 

\llT  J  0 


di. 


See  Art.  22. 
0  (/)  is  the  probability  that  the  error  will  be  less  nunierically 
than  the  limit  which  is  expressed  in  the  first  column  in  terms 
of  the  p.  e.  Thus,  the  sixth  line  of  the  table  means  that  out 
of  every  thousand  errors  the  chances  are  that  264  will  be  less 
than  one-half  as  great  as  the  p.  e. 

The  probability  that  an  error  a  is  greater  than  r  is  0.5,  than 
2  r  is  0.177,  than  3  r  is  0.043,  than  4  r  is  0.007,  than  5  r  is 
0.00 1,  than  6  r  is  0.000 1. 

TABLE  I. 
See  Art    2,5;. 


a 

@{i) 

Difference. 

a 

©W 

DlFFERENCE- 

r 

r 

0.0 

0.000 

2-5 

0.908 

O.I 

0.054 

54 

2 

6 

0.921 

13 

0.2 

0.107 

53 

2 

7 

0.931 

10 

0.3 

0.  160 

53 

2 

8 

0.941 

10 

0.4 

0.213 

53 

2 

9 

0.950 

9 

0.5 

0.264 

51 

3 

0 

0-957 

7 

0.6 

0-314 

50 

3 

I 

0 .  963 

6 

0.7 

0-363 

49 

3 

2 

0 .  969 

6 

0.8 

O.41I 

48 

3 

3 

0.974 

5 

0.9 

0.456 

45 

3 

4 

0.978 

4 

1 .0 

0.500 

44 

3 

5 

0.982 

4 

1 .1 

0.542 

42 

3 

6 

0.985 

3 

1 .2 

0.582 

40 

3 

7 

0.987 

2 

1-3 

0.619 

37 

3 

8 

0 .  990 

3 

1.4 

0-655 

36 

3 

9 

0.991 

I 

1-5 

0.688 

ZZ 

4 

0 

0-993 

2 

1.6 

0.719 

31 

4 

I 

0 .  994 

I 

1-7 

0.748 

29 

4 

2 

0.905 

I 

1.8 

0-775 

27 

4 

3 

0 .  996 

1 

1.9 

0.800 

25 

4 

4 

0.997 

1 

2.0 

0-823 

23 

4 

5 

0 .  998 

1 

2.1 

0.843 

20 

4 

6 

0.998 

0 

2.2 

0.862 

19 

4 

7 

0 . 9()8 

0 

2-3 

0.879 

17 

4 

8 

O.Q99 

I 

2.4 

0.895 

16 

•1 

0 

0.999 

0 

2-.^ 

0  .  908 

13 

5.0 

0. ()()() 

0 

292 


THE    ADJUSTMENT    OF    OBSERVATIONS 


TABLE   II. 
214.    Factors  for  BcsscV s  Probable  Error  Formulas. 
See  Art.  33. 


« 

_.674S_ 
\/« —  I 

.6745 

« 

.6745 

.6745 

\/ti  (»  —  i) 

V"  («  —  >) 

40 

oiioSo 

0.0171 

41 

.1066 

0.0167 

2 

0.6745 

0.4769 

42 

■1053 

0.0163 

3 

.4769 

•2754 

43 

.1041 

0.0159 

4 

•3894 

.1947 

44 

.  1029 

0.0155 

5 

0.3372 

0.1508 

45 

0.1017 

0.0152 

6 

.3016 

.1231 

46 

•  1005 

.014S 

7 

•2754 

.1041 

47 

.0994 

.0145 

8 

•2549 

.0901 

48 

.0984 

.0142 

9 

•2385 

■0795 

49 

.0974 

.0139 

10 

0.2248 

0.0711 

50 

0.0964 

0.0136 

II 

•2133 

.0643 

51 

.0954 

.0134 

12 

.2029 

•0587 

52 

.0944 

.0131 

13 

.1947 

.0540 

53 

•0935 

.0128 

14 

.1871 

.0500 

54 

.0926 

.0126 

IS 

0. 1803 

0.0465 

55 

0.0918 

0.0124 

16 

.1742 

•0435 

56 

.0909 

.0122 

17 

.1686 

.0409 

57 

.0901 

.0119 

18 

.1636 

.0386 

58 

.0893 

.0117 

19 

.1590 

•0365 

59 

.0886 

•0115 

20 

0.1547 

0.0346 

60 

0.0878 

0.0113 

21 

.1508 

.0329 

61 

.0871 

.0111 

22 

.1472 

.0314 

62 

.0864 

.0110 

23 

.1438 

.0300 

63 

■o8s7 

.0108 

24 

.  1406 

.02S7 

64 

.0850 

.0106 

25 

0-1377 

0.0275 

65 

0.0S43 

0.0105 

26 

•i'349 

.0265 

66 

.0837 

.0103 

27 

■-^i^?, 

•0255 

67 

.0830 

.0101 

28 

.1208 

.0245 

68 

.0824 

.0100 

29 

•1275 

•0237 

69 

.0818 

.0098 

30 

0.1252 

0.0229 

70 

0.0812 

0.0097 

31 

.1231 

.0221 

71 

.0806 

.0096 

32 

.1211 

.0214 

72 

.0800 

.  0094 

33 

.1192 

.0208 

73 

•0795 

.0093 

34 

.1174 

.0201 

74 

.0789 

.0092 

35 

0.1157 

0.0196 

75 

0.0784 

0.0091 

36 

.1140 

.OIQO 

80 

•0759 

.0085 

37 

.1124 

.0185 

85 

.0736 

.0080 

38 

.1109 

.oiSo 

90 

•0713 

.0075 

3, 

.1094 

•0175 

100 

.0678 

.0068 

APPENDIX 


293 


TABLE   III. 
215.    Factors  for  Peters    Probable  Error  Formulas. 

See  Art.  33. 


n 

.845J 

.S453 

n 

•8453 

•8453 
«  \/«  —  I 

V"  («  —  0 

«\/«  —  I 

\/«  (»  —  •) 

40 

0.0214 

0.0034 

. 

4t 

.0209 

•0033 

2 

0.5978 

0.4227 

42 

.0204 

.0031 

3 

■3451 

•1993 

43 

.0199 

.0030 

4 

.2440 

.1220 

44 

.0194 

.0029 

5 

0. 1890 

0.0845 

45 

0.0190 

0.0028 

6 

■1543 

.0630 

46 

.0186 

.0027 

7 

.1304 

•  0493 

47   ■ 

.0182 

.0027 

8 

.1130 

■0399 

48 

.0178 

.0026 

9 

.0996 

•0332 

49 

.0174 

.0025 

10 

0.0891 

0.0282 

50 

O.OI71 

0.0024 

II 

.0806 

•  0243 

51 

.0167 

•0023 

12 

.0736 

.0212 

52 

.0164 

.0023 

13 

.0677 

.0188 

53 

.0161 

.0022 

14 

.0627 

.0167 

54 

•0158 

.0022 

15 

0.0583 

0.0151 

55 

0.0155 

0.0021 

16 

.0546 

.0136 

56 

.0152 

.0020 

17 

•  0513 

.0124 

57 

.0150 

.0020 

18 

.0483 

.0114 

58 

•0147 

.0019 

19 

•0457 

.0105 

59 

.0145 

.0019 

20 

0.0434 

0.0097 

60 

0.0142 

0.0018 

21 

.0412 

.0090 

61 

.0140 

.0018 

22 

•0393 

.0084 

62 

•0137 

.0017 

23 

.0376 

.0078 

63 

•0135 

.0017 

24 

.0360 

.0073 

64 

•0133 

.0017 

25 

0-0345 

0 . 0069 

65 

0  .  0 1  3  I 

0.0016 

26 

•  0332 

.0065 

66 

.0129 

.  00 1 6 

27 

.0319 

.0061 

67 

.0127 

.0016 

28 

•0307 

.0058 

68 

.0125 

.0015 

29 

.0297 

•0055 

69 

•0123 

•0015 

30 

0.0287 

0.0052 

70 

0.0122 

0 . 00 1 5 

31 

.0277 

.0050 

71 

.0120 

.0014 

32 

.0268 

.0047 

72 

.Oil  8 

.0014 

33 

.0260 

.0045 

73 

.01 17 

.0014 

34 

.0252 

■0043 

74 

•  0 1 1 5 

•  00 1 3 

35 

0.0245 

0.0041 

75 

0 . 0 1 1 3 

0.0013 

36 

.0238 

.0040 

80 

.0106 

.0012 

37 

.0232 

.0038 

85 

.0100 

.00 II 

38 

30 

.0225 

.0220 

•  0037 
.0035 

90 

TOO 

.oo()5 

.00S5 

.0010 
.oooS 

INDEX 


The  figure  refers  to  the  page 


Accidental  error,  nature  of,  90,  273. 

Accuracy  not  limited  to  what  can  be 
seen,  48. 

Adjustment,  figure,  263. 

Adjustment,  general,  188. 

Adjustment,  local,  1S5. 

Adjustment  of  a  central  polygon,  234. 

Adjustment  of  a  level  net,  208. 

Adjustment  of  a  quadrilateral,  206, 
228,  231. 

Adjustment  of  triangulation,  method 
of  angles,  180. 

Adjustment  of  triangulation,  method 
of  directions,  230. 

Angle  equations,  189,  191. 

Angle  equations,  selection  of,  191. 

Approximate  method  of  finding  pre- 
cision, 237. 

Arithmetic  mean,  9,  35. 

Artifices,  two  special,  144. 

Assignment  of  weight  arbitrary,  82. 

Assignment  of  weights  in  a  level 
net,  270. 

Average  error,  24. 

Average  ratio. of  weights,  142. 

Azimuth  condition  equations,  253. 

Hase-line  measurement,  precision  of, 

261. 
Bessel's  formula  for  probable  error, 

38. 

Best  side  equations,  247. 

Blunders,  8. 

Bowditch's  rule  for  balancing  a  sur- 
vey, 158. 

Breaking  a  net  into  sections,  259. 

Caution  about  tests  of  precision,  45. 
Classification  of  observations,  34. 


Comparison  of  average,  mean  square 
and  probable  errors,  26. 

Comparison  of  observation  and 
theory,  45. 

Computation  of  normal  equations, 
132. 

Computation  of  [v"-],  32. 

Computing  machines,  loi. 

Condition  equations  for  length,  azi- 
muth, latitude,  and  longitude,  250. 

Conditions,  general,  number  of,  202. 

Conditions,  local,  number  of,  202. 

Conditions,  number  of,  202. 

Conditioned  observations,  169. 

Constant  error,  detection  of,  276. 

Constant  error  present,   weighting, 

77- 

Constant  errors,  51,  273. 

Control  of  [7/^],  42. 

Control  of  arithmetic  mean,  36. 

Control  of  formation  of  normal 
equations,  roo. 

Control  of  solution  of  normal  equa- 
tions, 107. 

Control  of  weighted  mean,  54. 

Control  of  weighted  mean,  57. 

Correlates,  method  of,  152,  210. 

Correlate  equations,  method  of  di- 
rections, 246. 

Curve  of  probability,  27. 

Detection  of  systematic  or  constant 

error,  276. 
Direct  observations,  one   unknown, 

35- 

Direction  method  of  adjustment  of 
triangulation,  iSo. 

Direction  method  of  observing  an- 
gles, 289. 


295 


296 


INDEX 


Distinction  between  accidental,  con- 
stant, and  systematic  errors,  273. 

Doolittle  method  of  solving  normal 
equations,  114. 

Elemental  errors  in  latitude  observa- 
tions, 279. 

Error,  accidental,  nature  of,  9,  273. 

Error,  average,  24. 

Error,  effect  of  extending  limits  of, 
3°- 

Error,  law  of,  13. 

Error,  mean  square,  22. 

Error  of  a  given  magnitude,  proba- 
bility of,  15. 

Error,  probable,  274. 

Error,  reduction  of,  by  repetition  of 
observations,  47. 

Error,  systematic,  274. 

Errors,  distinction  between  acci- 
dental,  constant,  and   systematic, 

273- 
Errors  distinguished  from  residuals, 

12. 
Errors,  instrumental,  274. 
Errors,  observer's,  5. 
Experience,  law  of  error  tested  by, 

44. 
External  conditions,  4. 

Factors  for  Bessel's  probable  error 
formula,  292. 

Factors  for  Peters'  probable  error 
formulas,  293. 

Figure  adjustment,  263. 

Formation  of  normal  equations,  96. 

Function  of  adjusted  values,  preci- 
sion of,  137. 

General  adjustment,  188. 
Groups,  solution  by,  69,  213,  223. 

Hagen's  hypothesis,  16. 

Iced-bar  measurements,  2S6. 
Improvement  in  methods  of  observa- 
tion, 273.  — 


Independent  angles,  methods  of,  180. 
Independent  angles,  method  of  ob- 
servation, 183. 
Indirect  observations,  93. 
Instruments,  i. 

Latitude  and  longitude  condition 
equations,  255. 

Latitude  observations,  zenith  tele- 
scope, 278. 

Law  of  error  tested  by  experience,  44. 

Law  of  error,  13. 

Least  squares,  principle  of,  19. 

Length  condition  equations,  52. 

Level  net,  adjustment  of,  268. 

Level  net,  assignment  of  weights, 
270. 

Limit  of  accuracy  not  the  limit  of 
vision,  48. 

Linear  function,  law  of,  error  of,  20. 

Linear  function,  precision  of,  137. 

Local  adjustment,  185. 

Local  conditions,  number  of,  188. 

Logarithmic  solution  of  normal 
equations,  112. 

Longitude  condition  equations,  250. 

Mean,  arithmetic,  9,  35. 

Mean,  weighted,  54. 

Mean  square  error,  22. 

Mean  square  error  7's.  probable  error, 

20. 
Methods  of  computing  [?'"],  183. 
Methods  of  observation,  selection  of, 

272. 
Multiples  of  the  unknown,  observed, 

60. 

Normal  equations,  98. 

Normal  equations,  forms  of  compu- 
ting, lOI. 

Normal  equations,  method  of  corre- 
lates, 152,  21C. 

Normal  equations,  solution  of,   105, 

177. 
Number  of  angle  equations,  189, 191. 


INDEX 


J97 


Number  of  general  conditions,  202. 
Number  of  local  conditions.  202. 
Number  of  side  equations,  193. 

Observations,  classification  of,  34. 
Observations,  conditioned,  149. 
Observations,  indirect,  93. 
Observations,  weighting  of.  94. 
Observed   values,    multiples  of   the 

unknown,  60. 
Observer's  errors.  5. 
Observing      angles      by      direction 

method,  239. 
Observing  by  method  of  independent 

angles,  183. 
One  unknown,  direct  observations, 

35- 

Personal  equation,  6. 

Peters'  formula  for  probable  error, 
40,  293. 

Pole,  position  of,  195,  201. 

Precision,  approximate  method  of 
finding,  237. 

Precisionof  adjusted  values,  121,  15S, 
162,  208,  211,  218. 

Precision  of  arithmetic  mean,  38. 

Precision  of  base-line  measurements, 
261. 

Precision  of  function  of  adjusted 
values,  137. 

Precision  of  a  linear  function,  62. 

Precision,  measure  of,  16. 

Precision  of  weighted  mean,  58. 

Predicted  tides,  2S9. 

Principle  of  least  squares,  19. 

Probability  curve,  27. 

Probability  of  error  of  a  given  mag- 
nitude, 32. 

Probable  error,  22. 

Probable  error  formula,  factors  for, 
292,  293. 

Probable  error,  independent  of  con- 
stant error,  46. 

Probable  error  of  single  observation, 
132. 


Probable  error  7'i-.mean  square  error, 

20. 
Probable  error,  more  accurate  deli- 

nition  of,  274. 

Quadrilateral,  adjustment  of,  206, 
228,  231. 

Ratio  of  weight  of  observed  to  ad- 
justed value,  143. 

Rejection  of  obsei-vations,  87. 

Relation  of,  probable  error  to  aver- 
age of  errors,  43. 

Residuals  distinguished  from  errors, 
12. 

Residuals,  squares   of,  a  minimi:m, 

13- 
Residuals,  sum  of  =  zero,  12. 
Repetition  of  observations  to  reduce 

error,  22. 

Sections,  breaking  a  net  into,  259. 

Selection  of  methods  of  observation, 
272. 

Selection  of  side  and  angle  condition 
equations,  189,  191- 

Side  equation,  reduction  to  linear 
form,  197. 

Side  equations,  193,  243. 

Side  equations,  best,  247. 

Side  equations,  number  of,  202. 

Single  observation,  probable  error 
of,  132. 

Solution  by  groups,  169,  213,  223. 

Solution  of  normal  equations,  105. 

Solution  by  successive  approxima- 
tion, 177. 

Squares  of  residuals  a  minimum,  19. 

St.  Gothard  tunnel,  221. 

Substitution,  method  of,  106. 

Summation,  symbol  of,  10. 

Telegraphic  longitude  observations, 

283. 
Tests  of  precision,  caution,  45. 
Time  of  solving  a    set  of   normal 

equations,  120. 


298 


INDEX 


Triangulation,  adjustment  of,  method 

of  angles,  180. 
Triangulation,  direction  method  of 

adjustment,  239. 
Trigonometric  leveling,  289. 

Weights,  55. 

Weights  arbitrarily  assigned,  82. 

Weights  in  a  level  net,  assignment 

of,  270. 
Weights  of  unknowns,  124,  129. 
Weighted  mean,  54. 
Weighted  mean,  examples  of,  71- 


Weighting  an  approximate  method 
of,  76. 

Weighting  a  function  of  knowledge, 
84. 

Weighting  when  constant  error  is 
present,  77- 

Weighting  of  observations,  74. 

Wright's  rule  for  adjusting  a  quad- 
rilateral, 229. 

Zenith  telescope  latitude  observa- 
tions, 278. 


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